![Preface
The objective of this book is to propose a modern introduction to the representa-
tion theory of the symmetric groups. There is now a large literature on the gen-
eral representation theory of finite groups, see for instance the classical Linear
Representations of Finite Groups by J.-P
. Serre ([Ser77]); and among this litera-
ture, a few books are concentrated on the case of symmetric groups, for example
The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric
Functions by B. Sagan (see [Sag01]). The point of view and interest of the present
book is the following: we shall show that most of the calculations on symmetric
groups can be performed, or at least eased by using some appropriate algebras of
functions. It is well known since the works of Frobenius and Schur that the alge-
bra of symmetric functions encodes most of the theory of characters of symmetric
groups. In this book, we shall use the algebra of symmetric functions as the start-
ing point of the representation theory of symmetric groups, and then go forward
by introducing other interesting algebras, such as:
• the algebra of observables of partitions, originally called “polynomial functions
on Young diagrams,” and whose construction is due to Kerov and Olshanski.
• the Hopf algebras of non-commutative symmetric functions, quasi-symmetric
functions and free quasi-symmetric functions, which contain and generalize the
algebra of symmetric functions.
This algebraic approach to the representation theory of symmetric groups can
be opposed to a more traditional approach which is of combinatorial nature, and
which gives a large role to the famous Young tableaux. The approach with algebras
of functions has several advantages:
1. First, if one tries to replace the symmetric group by finite-dimensional alge-
bras related to it (the so-called partition algebras, or the Hecke algebras),
then one can still use the algebra of symmetric functions to treat the charac-
ter theory of these algebras, and in this setting, most of the results related to
the symmetric groups have direct analogues. In this book, we shall treat the
case of Hecke algebras, which is a good example of this kind of extension of
the theory of symmetric groups (the case of partition algebras is treated for
instance in a recent book by Ceccherini-Silberstein, Scarabotti and Tolli, see
[CSST10]).
2. On the other hand, the algebraic approach leads to a new formula for the
irreducible characters of the symmetric groups, due to Stanley and Féray. The
xi](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-16-320.jpg)
![4 Representations of finite groups and semisimple algebras
(G1) ·G is associative and admits a neutral element:
∀g,h, i ∈ G, (g ·G h) ·G i = g ·G (h ·G i);
∃e ∈ G, ∀g ∈ G, g ·G e = e ·G g = g.
The neutral element e is then unique.
(G2) every element of G has a (unique) inverse for the product:
∀g ∈ G, ∃h ∈ G, g ·G h = h ·G g = e.
We shall usually omit the notation ·G, and just denote g·G h = gh. Also, the inverse
of g ∈ G will be denoted g−1
, and the neutral element will be denoted indiffer-
ently eG, e or 1. A group will be called finite if as a set it has finite cardinality. We
then write
|G| = card G = number of elements of |G|.
Example. Let p be a prime number, and denote Z/pZ = {[1],[2],...,[p]} the set
of classes of integers modulo p, which is a ring (quotient of the ring of integers
Z). Endowed with the product of classes
[a] × [b] = [ab],
the set (Z/pZ)∗
= {[1],[2],...,[p −1]} is a finite group of cardinality p −1, with
neutral element e = [1]. It is commutative, which means that for any g,h ∈ G,
gh = hg.
Example. If S is a set, denote S(S) the set of maps σ : S → S that are bijective.
This is a group with respect to the operation of composition of maps; the neutral
element is the identity
idS : s ∈ S 7→ s,
and the inverse of a bijection σ ∈ S(S) is the inverse function σ−1
with t = σ(s)
if and only if s = σ−1
(t). This book is devoted to the study of the groups S(n) =
S([
[1, n]
]), where [
[1, n]
] is the set of integers {1,2,3,..., n − 1, n} between 1 and
n. We shall say a bit more about them in a moment.
A morphism between two groups G and H is a map φ : G → H compatible
with the products of G and H, i.e., such that
φ(g1 g2) = φ(g1)φ(g2)
for all g1, g2 ∈ G. One speaks of isomorphism of groups if φ is bijective; then, the
inverse map φ−1
is also a morphism of groups. On the other hand, a subgroup
H of a group G is a subset of G stable by the operations of product and inverse:
∀h1,h2 ∈ H, h1h2 ∈ H ; ∀h ∈ H, h−1
∈ H.
Then, H is a group for the restriction of the product map from G × G to H × H.
In the following, we shall say that a group H can be seen as a subgroup of G if
there is an injective morphism of groups φ : H → G, which thus identifies H with
a subgroup of G.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-25-320.jpg)
![Finite groups and their representations 5
Example. Let k be a field, and V be a k-vector space. Then GL(V), the set of
bijective linear maps φ : V → V, is a group for the operation of composition of
maps. Similarly, if k = C and V is a complex vector space endowed with a scalar
product, then the set U(V) of linear isometries of V is a group for the composition
of maps, and it is a subgroup of GL(V). Going to the matrix point of view, the
following sets of matrices are also groups for the product of matrices, the neutral
element being the identity matrix In = diag(1,1,...,1):
GL(n, k) = {M ∈ M(n, k) | det(M) 6= 0};
SL(n, k) = {M ∈ M(n, k) | det(M) = 1};
U(n,C) = {M ∈ M(n,C) | M∗
M = M M∗
= In}.
If V is a complex vector space of dimension n, then the groups of matrices
GL(n,C), SL(n,C) and U(n,C) are isomorphic respectively to GL(V), SL(V) and
U(V), the isomorphism being the map which sends a linear map to its matrix in
a (unitary) basis. On the other hand, for GL(n, k) to be finite, we need k to be
finite, and then, if q = card k, one has
cardGL(n, k) = (qn
− 1)(qn
− q)(qn
− q2
)...(qn
− qn−1
).
Indeed, this is the number of distinct bases of kn
to which an arbitrary basis
(e1,..., en) can be sent by an element of GL(n, k) = GL(kn
).
 Symmetric groups.
The symmetric group of order n is the group of bijections S(n) = S([
[1, n]
]). It
is a finite group with cardinality
|S(n)| = n! = 1 × 2 × 3 × ··· × n =
n
Y
i=1
i.
Indeed, to choose a bijection σ between elements of [
[1, n]
], one has:
• n possibilities for the image σ(1) of 1 (all the integers between 1 and n);
• n−1 possibilities for the image σ(2) of 2 (all the integers but the one already
chosen for σ(1));
• in general, assuming the images σ(1),...,σ(k) already chosen, n − k possi-
bilities for the image σ(k + 1) of k + 1.
Multiplying these choices yields |S(n)| = n!. The elements of S(n) are called per-
mutations of size n, and we shall denote σ = σ(1)σ(2)...σ(n) a permutation
given by the list of its values. So for instance, 4132 is the permutation in S(4)
that sends 1 to 4, 2 to 1, 3 to 3 and 4 to 2.
Let us now list some easy and well-known properties of these symmetric
groups:](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-26-320.jpg)
![6 Representations of finite groups and semisimple algebras
1. If n ≤ N, then S(n) can be seen naturally as a subgroup of S(N). Indeed,
a bijection σ between the n first integers can be extended in a bijection e
σ
between the N first integers by setting:
e
σ(k) =
¨
σ(k) if k ≤ n,
k if n < k ≤ N.
In the sequel, we shall use these natural imbeddings S(n) ,→ S(N) constantly,
and unless the distinction is needed, we shall keep the same notation for a
bijection σ ∈ S(n) and its extension to a larger symmetric group S(N).
2. For n ≥ 3, S(n) is a non-commutative group, which means that one can find
g and h such that gh 6= hg. Indeed, using the previous property, it suffices to
prove the case n = 3, and in this case, if σ = 321 and τ = 213, then their
composition products are
στ = 231 ; τσ = 312
and they are different.
3. Any finite group can be seen as a subgroup of a finite symmetric group. Indeed,
given a finite group G, consider the map
φ : G → S(G)
g 7→ (φ(g) : h 7→ gh).
This is a morphism of groups, which is injective since g can be recovered from
the map φ(g) by the formula g = φ(g)(e). On the other hand, given two finite
sets A and B with the same cardinality and a bijection ψ : A → B, there is an
isomorphism between the groups S(A) and S(B), namely,
Ψ : S(A) → S(B)
σ 7→ ψ ◦ σ ◦ ψ−1
.
Thus, if n = card G, then S(G) and S(n) are isomorphic, so G can be seen as
a subgroup of S(n).
A more crucial property of permutations deserves the following proposition.
Call cycle of length k and support (a1, a2,..., ak) the permutation that sends a1
to a2, a2 to a3, a3 to a4, etc., and ak to a1; and that leaves invariant all the other
elements of [
[1, n]
]. For instance, the cycle (1,4,2) in S(4) sends 1 to 4, 4 to 2, 2
to 1, and the remaining element 3 to itself; thus, (1,4,2) = 4132. If k ≥ 2, then a
cycle of length k and support (a1,..., ak) is uniquely determined by the sequence
(a1, a2,..., ak), up to a cyclic permutation of this sequence:
(a1, a2,..., ak) = (a2, a3,..., ak, a1) = (a3,..., ak, a1, a2) = ···
On the other hand, a cycle of length 1 is just the identity permutation, and can
be seen as a way to design a particular fixed point (later, we shall make this idea
of marked fixed point more rigorous with the notion of partial permutation).](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-27-320.jpg)
![Finite groups and their representations 7
Proposition 1.1. Any permutation σ ∈ S(n) can be written as a product of cycles
with disjoint supports, the sum of the lengths of these cycles being equal to n. This
decomposition
σ = c1 ◦ c2 ◦ ··· ◦ cr
is unique up to permutation of the cycles c1,..., cr .
Proof. In a finite group, every element g has for inverse a positive power of itself
gk≥1
. As a consequence, the permutation σ ∈ S(n) being fixed, the relation on
[
[1, n]
] defined by
i ∼σ j ⇐⇒ ∃k ≥ 0, j = σk
(i),
which is clearly reflexive and transitive, is also symmetric, so it is an equivalence
relation. Call orbit of σ a class for the equivalence relation ∼σ on [
[1, n]
]; then,
the orbit of i, if it has length k, is {i,σ(i),σ2
(i),...,σk−1
(i)}, and the restriction
of σ to this orbit is the cycle
c = (i,σ(i),σ2
(i),...,σk−1
(i)).
The decomposition of σ in disjoint cycles is then obtained by choosing one
representative for each orbit, and the unicity comes from the fact that if σ =
c1 ◦ c2 ◦ ··· ◦ cr is a product of cycles with disjoint supports, then these supports
are orbits of σ, with the order of elements for each cycle entirely determined by
the action of σ on each support.
Example. Consider the permutation σ = 874312659 in S(9). Its orbits are
{1,5,8}, {2,6,7}, {3,4} and {9}, and the cycle decomposition of σ is
σ = (1,8,5)(2,7,6)(3,4)(9).
Thus, we get two different writings for a given permutation σ ∈ S(n): the
notation in line σ = σ(1)σ(2)...σ(n), and the cycle decomposition
σ = (a1,..., ar )(b1,..., bs)···(z1,...,zt ).
In the cycle decomposition, it will sometimes be convenient to omit the cycles of
length 1, since they correspond to the identity. This is in particular the case if σ
is itself a single cycle. We say that σ is a transposition if it is a cycle of length 2;
then, it writes as σ = (i, j) = (j, i) and it exchanges i and j.
 Representations of groups.
If V is a complex vector space, we denote as before GL(V) the group of complex
linear isomorphisms u : V → V. If V is finite-dimensional and if (e1,..., en) is a
fixed linear basis of V, we denote (ui j)1≤i,j≤n the matrix of the linear map u in
this basis, which means that
u(ej) =
n
X
i=1
ui j ei.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-28-320.jpg)
![Characters and constructions on representations 15
Both sides of the equation are linear forms in the coefficients of u, so their
coefficients must be equal. Therefore, for any coefficients i, j ∈ [
[1, dλ]
] and
k, l ∈ [
[1, dµ]
],
¬
ρλ
i j ρ
µ
lk
¶
G
=
1
|G|
X
g∈G
ρλ
i j(g)ρ
µ
lk
(g)
=
1
|G|
X
g∈G
ρλ
i j(g−1
)ρ
µ
kl
(g) =
¨δi.l δj,k
dim Vλ if λ = µ,
0 otherwise.
This is the equality stated by the theorem with the roles of k and l exchanged. By
taking i = j and k = l, and summing over indices i ∈ [
[1, dλ]
] and k ∈ [
[1, dµ]
], we
get the orthogonality relation for characters.
Corollary 1.9. A representation V of a finite group G is entirely determined (up to
isomorphisms) by its character.
Proof. If V =
L
λ∈b
G mλ Vλ
, then chV
=
P
λ∈b
G mλ chλ
. By using the orthogonality
of characters, one sees then that the multiplicity mλ of an irreducible represen-
tation Vλ
in V is given by mλ = chV
chλ
G
. As a by-product, one gets a cri-
terion of irreducibility for a representation of G: V is irreducible if and only if
chV
chV
G
= 1.
Example. Consider the permutation representation of S(n) on Cn
, with n ≥ 2. Its
character is
chCn
(σ) =
n
X
i=1
δi,σ(i) = number of fixed points of σ.
Let us compute the norm of this character:
chCn
chCn
S(n)
=
1
n!
X
σ∈S(n)
‚ n
X
i=1
δi,σ(i)
Œ2
=
1
n!
n
X
i,j=1
X
σ∈S(n)
δi,σ(i) δj,σ(j)
!
=
1
n!
n
X
i=1
(n − 1)! +
X
1≤i6=j≤n
(n − 2)!
!
=
n! + n!
n!
= 2.
Indeed, on the second line, if two indices i and j are fixed, then the number of
permutations with σ(i) = i and σ(j) = j is (n − 2)!; and similarly, if one index i
is fixed, then the number of permutations with σ(i) = i is (n − 1)!.
It follows that Cn
is necessarily the direct sum of 2 non-isomorphic irreducible
representations; indeed, if V =
L
λ∈b
G mλ Vλ
, then chV
chV
G
=
P
λ∈b
G (mλ)2
. By
the discussion on page 10, these two components are the trivial representation on
C(1,1,...,1), and the representation on {(x1,..., xn) ∈ Cn
| x1 + ··· + xn = 0}.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-36-320.jpg)
 =
|G|
|H|
dim V.
We make G act on IndG
H (V) by (g · f )(g0
) = f (g0
g), and we call IndG
H (V) the
induced representation of V from H to G.
Proposition 1.11. The character of an induced representation IndG
H (V) is given by
the formula
chIndG
H (V)
(g) =
X
gj ∈G/H
chV
(g−1
j ggj),
where chV
(g) = 0 if g is not in H, and the sum runs over representatives g1,..., gr
of the left cosets gH.
We postpone the proof of this proposition to the end of Section 1.4, where the
framework of algebras and modules will provide a more natural definition of
the induction of representations, and a simple explanation of the formula for
characters.
 Frobenius’ reciprocity.
An important feature of the operations of induction and restriction of representa-
tions is their adjointness in the sense of functors on categories. More concretely,
one has:
Proposition 1.12 (Frobenius). Let H ⊂ G be two finite groups, and V and W be
two representations of H and G. One has
¬
chV
chResG
H (W)
¶
H
=
¬
chIndG
H (V)
chW
¶
G
.
In particular, if V and W are irreducible, then the multiplicity of V in ResG
H (W) is
the same as the multiplicity of W in IndG
H (V).](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-38-320.jpg)
![22 Representations of finite groups and semisimple algebras
and we keep the same Hilbert scalar product on CG as before. We define as before
the dual group algebra Ce
G =
L
λ∈e
G End(Vλ
), but we change the scalar product
of Ce
G into
〈u| v〉e
G =
X
λ∈e
G
dλ tr((uλ
)∗
vλ
).
We define the Fourier transform of a function f by
e
f (λ) =
1
|G|
X
g∈G
f (g)ρλ
(g).
Then, the new Fourier transform f 7→ e
f is as before an isomorphism of C-algebras,
of G-representations, and of Hilbert spaces. Moreover:
Proposition 1.15. For any function f on the group G,
f (g) =
X
λ∈e
G
dλ tr ρλ∗
(g) e
f (λ)
;
〈f | f 〉G = e
f e
f e
G
=
X
λ∈e
G
dλ tr (e
f (λ))∗ e
f (λ)
.
These formulas are exactly the same as those satisfied by the Fourier series of a
square-integrable function f on the circle T = R/(2πZ):
f (θ) =
X
k∈Z
e
f (k)e−ikθ
;
1
2π
Z 2π
0
|f (θ)|2
dθ =
X
k∈Z
e
f (k)
2
where e
f (k) = 1
2π
R 2π
0
f (θ)eikθ
dθ. The reason for this correspondence is that the
formulas of Proposition 1.15 hold in fact for any square-integrable function on a
topological compact group, the means
1
|G|
X
g∈G
·
being replaced in this theory by integrals
Z
G
· Haar(dg)
against the Haar measure. For instance, with the circle T, the set of irreducible
representations is labeled by Z, each irreducible representation of label k ∈ Z be-
ing one-dimensional and given by ρ(θ) = eikθ
. The Haar measure on T = [0,2π]
is dθ
2π and Proposition 1.15 gives indeed the Fourier series of harmonic functions.
We leave as an exercise to the reader (see also the notes at the end of the chapter)
the proof that almost all results proved so far for representations and characters of
finite groups extend to topological compact groups, the only difference being that](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-43-320.jpg)
![The non-commutative Fourier transform 25
Proof. Since irreducible characters form an orthonormal basis of Z(CG),
f (g) =
X
λ∈b
G
chλ
f G
chλ
(g) =
X
λ∈Ĝ
(dλ)2
χλ
f G
χλ
(g).
Then, χλ
f G
= 1
|G|
P
g∈G f (g)χλ
(g−1
) = 1
|G|
P
g∈G f ∗
(g)χλ
(g) =
c
f ∗(λ)
|G| .
The last result involves the so-called Plancherel measure Pl(λ) =
(dλ)2
|G| . This
is a probability measure on b
G, and for any central function,
f =
Z
b
G
c
f ∗(λ) Pl(λ) chλ
.
In particular, consider the normalized character of the regular representation of
G:
χCG
(g) = δe,g.
It has for Fourier transform the constant function equal to 1, since
χCG
=
X
λ∈b
G
Pl(λ)χλ
.
Thus, the Plancherel measure corresponds to the decomposition in normalized
irreducible characters of the normalized regular trace of the group. The study of
this probability measure in the case of symmetric groups will be the main topic
of Chapter 13 of this book.
Example. As an application of the results of this section, consider the following
random process on the symmetric group S(n). We consider a deck of cards that
are ordered from 1 to n, and at each time k ∈ N, we choose at random two
independent indices i, j ∈ [
[1, n]
], and we exchange the i-th card of the deck with
the j-th card of the deck, cards being counted from top to bottom. Each index i or
j has probability 1
n , and it is understood that if i = j, then one leaves the deck of
cards invariant. The configuration after k random transpositions of cards can be
encoded by a permutation σk of size n, with σ(1) denoting the label of the first
card of the deck, σ(2) denoting the label of the second card of the deck, etc. For
instance, assuming n = 5, a possible trajectory of the process up to time k = 8 is
k 0 1 2 3 4 5 6 7 8
σk 12345 15342 13542 43512 43512] 43521 23514 25314 25314
and there are two steps (k = 4 and k = 8) where the same index i = j was chosen.
We denote P[A] the probability of an event A, and P[A| B] the probability of](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-46-320.jpg)
![26 Representations of finite groups and semisimple algebras
an event A conditionally to another event B. Consider the law fk(σ) = P[σk = σ],
viewed as an element on CS(n). The rules of the random process are:
P[σk+1 = τ|σk = σ]
=
¨
1
n if τ = σ (corresponding to choices of indices i = j);
2
n2 if τ = σ(i, j) for some pair i 6= j.
Therefore, we get a recursion formula for fk:
fk+1 =
X
τ∈S(n)
P[σk+1 = τ]τ =
X
σ,τ∈S(n)
fk(σ)P[σk+1 = τ|σk = σ]τ
=
X
σ∈S(n)
fk(σ)σ
1
n
+
1
n2
X
1≤i6=j≤n
(i, j)
!
= fk
1
n
+
1
n2
X
1≤i6=j≤n
(i, j)
!
.
So, if f = 1
n + 1
n2
P
1≤i6=j≤n(i, j), then fk = f k
for any k, in the sense of convolution
in CS(n). It should be noticed that the recursion formula can be rewritten as
fk+1 − fk = fk
1
n2
X
1≤i6=j≤n
(i, j) − 1
!
.
Thus, we are looking at the analogue in the setting of the symmetric group of the
heat equation
∂ f
∂ t = 1
2 ∆f .
Notice now that f is a linear combination of conjugacy classes in the symmet-
ric group. Indeed, the identity 1 is a conjugacy class on its own, and on the other
hand, two transpositions (i, j) and (k, l) are always conjugated:
(k, l) = (i, k)(j, l)(i, j)(j, l)−1
(i, k)−1
and each transposition appears twice in
P
1≤i6=j≤n(i, j). Moreover, since 1 = (1)−1
and (i, j) = (i, j)−1
, f ∗
= f . Therefore, f is in Z(CS(n)), and
fk(σ) =
X
λ∈Ò
S(n)
Pl(λ) Ö
(f k)∗(λ) χλ
(σ) =
X
λ∈Ò
S(n)
Pl(λ) b
f (λ)
k
χλ
(σ).
since the Fourier transform is an isomorphism of algebras. This can be rewritten
as:
fk(σ) =
X
λ∈Ò
S(n)
Pl(λ)
1
n
+
n − 1
n
χλ
(1,2)
‹k
χλ
(σ),
since χλ
(1) = 1 for any representation, and χλ
(i, j) = χλ
(1,2) for any transpo-
sition (i, j) and any representation λ. This formula can be used to compute the
asymptotics of the laws fk. In particular, it can be shown that −1 ≤ χλ
(1,2) 1 if
λ is not the trivial representation of S(n) on C. As a consequence, all the terms of](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-47-320.jpg)
![28 Representations of finite groups and semisimple algebras
(A1) associative and with a (unique) neutral element (the unity of the algebra):
∀a, b, c ∈ A, (a ×A b) ×A c = a ×A (b ×A c);
∃1A ∈ A, ∀a ∈ A, a ×A 1A = 1A ×A a = a.
(A2) compatible with the external product of k:
∀λ ∈ k, ∀a, b ∈ A, λ(a ×A b) = (λa) ×A b = a ×A (λb).
(A3) distributive with respect to the internal addition:
∀a, b, c ∈ A, a×A(b+c) = a×A b+a×Ac and (a+b)×Ac = a×Ac+b×Ac.
In other words, a k-algebra is a ring and a k-vector space whose structures are
compatible with one another in every possible way that one can think of. In the
two first parts of this book, unless explicitly stated, we shall only work with finite-
dimensional algebras, and denote dimk A, or simply dim A the dimension of A as a
k-vector space. An algebra is said to be commutative if its product ×A is commu-
tative: a ×A b = b ×A a. As before, we shall omit in most cases the product ×A and
write a ×A b = ab. The properties listed above for an algebra ensure that this is a
non-ambiguous notation.
Example. Given a finite-dimensional k-vector space V, the set End(V) of k-linear
maps u : V → V endowed with the product of composition of functions is a
finite-dimensional algebra of dimension (dim V)2
. Similarly, the set of matrices
M(n, k) of size n×n and with coefficients in k is an algebra for the matrix product.
If n = dimk V, then the two algebras End(V) and M(n, k) are isomorphic, an
isomorphism being given by
u 7→ mat(e1,...,en)(u),
where (e1,..., en) is an arbitrary basis of V.
Example. For any finite group G, the set kG of formal k-linear combinations of
elements of G (or, in other words, the set of functions G → k) is a k-algebra for
the convolution product defined at the beginning of Section 1.3. It has dimension
dimk kG = |G|.
Example. For any field k, the set k[X1,..., Xn] of polynomials in n variables with
coefficients in k is a commutative k-algebra. It is graded by the degree of polyno-
mials, and this gradation is compatible with the algebra structure, meaning that
for any elements a and b in the algebra,
deg(ab) ≤ deg a + deg b.
This inequality is an equality as soon as a and b are not zero.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-49-320.jpg)
![30 Representations of finite groups and semisimple algebras
left module A associated to an algebra A, the action A× A → A being given by the
product of the algebra. Then, in the second part of Lemma 1.5, the isomorphism
of vector spaces between V and HomG(CG, V) comes from the more general fact
that for any k-algebra A and any left A-module M, the module M is isomorphic
as a k-vector space to HomA(A, M), the isomorphism being
m 7→ (a 7→ a · m).
If M is a left A-module, then M is also canonically a right Aopp
-module, where
Aopp
denotes the k-algebra with the same underlying vector space as A, and with
product
a ×Aopp b = b ×A a.
The right Aopp
-module structure on a left A-module M is then defined by m·Aopp a =
a·A m. In the case where A = CG, there is a simple realization of Aopp
by using the
inverse map. More precisely,
CG → (CG)opp
X
g∈G
f (g) g 7→
X
g∈G
f (g) g−1
is an isomorphism of C-algebras. Therefore, any left representation V of G admits
a corresponding structure of right representation of G, given by
v · g = g−1
· v.
Example. The permutation representation of S(n) on Cn
is more natural when
given by a structure of right CS(n)-module. Indeed, it writes then as
(x1, x2,..., xn) · σ = (xσ(1), xσ(2),..., xσ(n)).
Similarly, consider a finite alphabet A, and denote the elements of An
as words of
length n with letters in A:
An
= {a1a2a3 ... an | ∀i ∈ [
[1, n]
], ai ∈ A}.
There is a natural structure of right CS(n)-module on the space C[An
] of formal
linear combinations of these words:
(a1a2a3 ... an) · σ = aσ(1)aσ(2)aσ(3) ... aσ(n).
There is also a natural structure of left CS(A)-module given by
τ · (a1a2a3 ... an) = τ(a1)τ(a2)τ(a3)...τ(an).
This kind of construction justifies the need of both notions of left and right mod-
ules on an algebra. We shall study this double action more rigorously in Section
1.5, by introducing the notion of bimodule.
Example. Let A be any k-algebra. Then A is both a left and right A-module, for
the actions given by the product of the algebra. The left A-submodules of A are
exactly the left ideals of A, and similarly on the right.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-51-320.jpg)
![Semisimple algebras and modules 31
 Semisimplicity and Artin–Wedderburn theorem.
Let A be a k-algebra, and M be a (left) module over A that is not the zero mod-
ule. The module M is said to be simple if it is of positive dimension and if its
only submodules are {0} and M itself. It is said to be semisimple if it is a direct
sum of simple modules. The notion of simple module is the generalization to the
framework of modules and algebras of the notion of irreducible representation,
and indeed, a representation V of G is irreducible if and only if it is a simple
CG-module.
Proposition 1.20. A finite-dimensional module M over a k-algebra A is semisimple
if and only if, for every submodule N ⊂ M, there exists a complement A-submodule
P with M = N ⊕ P.
Proof. Suppose that M =
Lr
i=1 Mi is a direct sum of simple modules, and let N be
a submodule of M. We take a subset I ⊂ [
[1, r]
] that is maximal among those such
that N ∩
L
i∈I Mi = {0}. By choice, if P =
L
i∈I Mi, then N + P is a direct sum. We
claim that N ⊕ P = M. It suffices to show that for every i ∈ [
[1, r]
], N ⊕ P contains
Mi. This is clear if i ∈ I. If i /
∈ I, then (N ⊕ P) ∩ Mi is not the zero submodule,
since otherwise the set I would not be maximal. But Mi is simple, so (N ⊕ P)∩ Mi
is a non-zero submodule of Mi, hence equal to the whole of Mi. This proves the
existence of a complement A-module P of N such that M = N ⊕ P.
Conversely, suppose that every submodule N of M has a complement sub-
module P. We can exclude the trivial case M = {0}. Then, since M is finite-
dimensional, there is no infinite descending chain of submodules of M, so M
has necessarily a simple submodule M1. Denote P a complement of M1:
M = M1 ⊕ P with M1 simple.
To show that M is semisimple, it suffices now to prove that P has the same prop-
erty as M, that is to say, that every submodule of P has a complement submodule
in P. Indeed, an induction on the dimension of M will then allow us to conclude.
Fix a submodule S ⊂ P. There is an isomorphism of A-modules
ψ : P → M/M1
p 7→ [p]M1
.
The A-submodule ψ(S) of M/M1 can be realized as the quotient module (M1 ⊕
S)/M1. By hypothesis, (M1 ⊕ S) has a complement R in M:
M = (M1 ⊕ S) ⊕ R.
Then, if πM1
: M → M/M1 is the canonical projection, πM1
(R) is a complement
submodule of ψ(S) in M/M1, and T = ψ−1
(πM1
(R)) is a complement submodule](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-52-320.jpg)
![though my baptism had been deep. I found that the greatest part of
the members of our Society and many of the ministers and elders,
are carried away by the principle which Elias Hicks has so
assiduously propagated among them. He now speaks out boldly,
disguising his sentiments no longer; he seeks to invalidate the Holy
Scriptures, and sets up man’s reason as his only guide, openly
denying the divinity of Christ. I have had many expostulations with
him in which I have most tenderly pleaded with him, but all has
been in vain.” [374] From what I have read in American literature
touching what is known as the Hicksite controversy, it appears to me
plainly indicative of a denial among many American Friends, that
Jesus Christ, in the orthodox sense of the term, was Divine, and that
He did not make any atonement for sin. Hicks appears to have been
a thorough mystic, unintelligible to common-sense people. At all
events he converted many to his views; and these views were
caught up by some Friends in this country. To what extent exactly
they were adopted in England I cannot say: but they created alarm
amongst many Friends on this side the Atlantic. Great sorrow at the
abandonment of Evangelical doctrines led to secessions from
Quakerism on the part of excellent people who had been born and
bred in the community. Some of them resided, at the time I speak
of, on the borders of Wales, others in the county of York. They
became Congregationalists, and in tours on behalf of the London
Missionary Society, I was received hospitably in their homes, and
had gratifying opportunities of witnessing their beautiful Christian
life.
Joseph John Gurney, of Earlham, felt seriously concerned respecting
the American defection, in a community to which he had been
attached from childhood. He had studied in the University of Oxford,
had cultivated friendships in other denominations, was a good classic
and Biblical scholar, and also an author of theological works. Mr.
Gurney was “concerned” about the effect of Hicksite opinion on
American and English Friends, and therefore took up his pen and
wrote in reply to the leader who had done so much mischief.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-60-320.jpg)
![Mr. Gurney, like his sister Mrs. Fry, undertook journeys for preaching
the Gospel, and once he visited Windsor for that purpose. I was
unwell at the time, but he called and talked by my bedside, and
commended me to God in prayer. Several Quaker families at that
period were living at Staines and Uxbridge; with them I had much
intercourse, especially when we were joined in the advocacy of Slave
Emancipation. The community, in both towns now named, was
considerable for numbers and for wealth.
Friends now dress, speak and act much like other people.
Conforming to common custom, they still eschew all extravagances
of fashion. They no longer forfeit membership by “marrying out of
Society.” “The Right Honourable John Bright” (how shocked George
Fox would have been at the title!) told me once, that relaxation in
strictness as to unimportant points, had checked a decline in
numbers going on before.
V. Methodism, of course, brings to my mind a long train of early
associations. Not merely names, but living forms, of noted
preachers belonging to the second decade of this century come back
to my recollection.
Calvert Street Chapel was opened about 1812, and Dr. Coke
preached.
I cannot say that I remember his sermon; but, as noticed already, I
distinctly recollect seeing the odd-looking, diminutive man, standing
on a table talking in the committee room of Bethel Hospital [377]
adorned by paintings of foundress and governors. Dr. Coke
energetically addressed on the occasion a number of people, who
had been invited by my grandfather, to hear the noted advocate of
Methodist missions. Many years afterwards I mentioned the
circumstance to a gentleman, who at the time took care of the
patients, when he fetched an old committee book, in which this
gathering was noticed, with a minute expressing the displeasure of
the Governors at such a liberty being taken, and forbidding anything
of the kind in future. The Wesleyan congregations in Norwich were](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-61-320.jpg)
![FOOTNOTES
[77] Faulkener’s “History and Antiquities of Kensington,” p. 317.
[78] 1893.
[80] “Christian Workers of the Nineteenth Century,” S.P.C.K., p. 216.
[88a] “Life of E. B. Pusey,” i. 336.
[88b] Ibid., ii. 33.
[89] “Life of Pusey,” ii. 8.
[126] Early Independent Churches had been particular in their
relations to one another; and they would not recognise new
communities without satisfactory evidence of character, principles,
and conduct. They became more isolated afterwards.
[176a] Now Archbishop of York.
[176b] A very good account of this under the title of “Lectures on
Bible Revision,” has been published by my excellent friend and late
colleague at New College, Principal Newth, D.D.
[183] “Memorials of a Quiet Life,” i. 237.
[184] Dr. Raleigh, Sir Charles Reed, and others, were examined.
[193] That was whilst I was in full work at Kensington, and not very
long after our new chapel was built, while a debt of £1000 rested on
it. I said I could not leave my charge whilst that debt remained. As
soon as I had declined the New College principalship, my](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-71-320.jpg)
![congregation swept off the debt as expressive of gratitude for my
remaining amongst them.
[197] “Ecce Homo,” chap. iv.
[230] Written about 1883.
[233] I am glad that at Kensington, a liturgical element has been
introduced, such as I should have approved, but could not
accomplish, because I knew it would then be disapproved by many.
[248] With a short Memoir by Robert Hall.
[250] In what I have ventured to say about pulpit preparation I
have hoped to help my younger ministerial brethren.
[252] “Homes and Haunts of Martin Luther,” p. 4.
[268] Since my visit to Ban de la Roche I discovered that, in a part
of the country not far off, an Irish missionary, Columbanus, in the
sixth century laboured for the temporal, as well as the spiritual,
welfare of the people. See Wolf’s “Country of the Vosges,” p. 214.
[315] Eusebius, “Eccl. Hist.,” V. i, 2.
[316] Pastor and Madame Rodriguez.
[318] De Aniccio, “L’Espagne traduit de Italien.”
[329] “Life of Wilkie,” p. 472.
[333] I have gone into this story in my “Spanish Reformers,” p. 185.
[374] “Memoirs of Stephen Grellet,” vol. ii., 130.
[377] See page 2.](https://image.slidesharecdn.com/2947255-250520060320-67b35515/85/Representation-Theory-Of-Symmetric-Groups-1st-Edition-Mliot-72-320.jpg)
Representation Theory Of Symmetric Groups 1st Edition Mliot
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- 6. Representation Theory of Symmetric Groups
- 7. DISCRETE MATHEMATICS ITS APPLICATIONS R. B. J. T. Allenby and Alan Slomson, How to Count: An Introduction to Combinatorics, Third Edition Craig P. Bauer, Secret History: The Story of Cryptology Jürgen Bierbrauer, Introduction to Coding Theory, Second Edition Katalin Bimbó, Combinatory Logic: Pure, Applied and Typed Katalin Bimbó, Proof Theory: Sequent Calculi and Related Formalisms Donald Bindner and Martin Erickson, A Student’s Guide to the Study, Practice, and Tools of Modern Mathematics Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Miklós Bóna, Combinatorics of Permutations, Second Edition Miklós Bóna, Handbook of Enumerative Combinatorics Miklós Bóna, Introduction to Enumerative and Analytic Combinatorics, Second Edition Jason I. Brown, Discrete Structures and Their Interactions Richard A. Brualdi and Dragos̆ Cvetković, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Abhijit Das, Computational Number Theory Matthias Dehmer and Frank Emmert-Streib, Quantitative Graph Theory: Mathematical Foundations and Applications Martin Erickson, Pearls of Discrete Mathematics Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence
- 8. Titles (continued) Mark S. Gockenbach, Finite-Dimensional Linear Algebra Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross, Jay Yellen, and Ping Zhang Handbook of Graph Theory, Second Edition David S. Gunderson, Handbook of Mathematical Induction: Theory and Applications Richard Hammack, Wilfried Imrich, and Sandi Klavžar, Handbook of Product Graphs, Second Edition Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Darel W. Hardy, Fred Richman, and Carol L. Walker, Applied Algebra: Codes, Ciphers, and Discrete Algorithms, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words Leslie Hogben, Handbook of Linear Algebra, Second Edition Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Neil P. Sigmon, and Ernest L. Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB® , Second Edition Richard E. Klima and Neil P. Sigmon, Cryptology: Classical and Modern with Maplets Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William L. Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization, Second Edition Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Hang T. Lau, A Java Library of Graph Algorithms and Optimization C. C. Lindner and C. A. Rodger, Design Theory, Second Edition San Ling, Huaxiong Wang, and Chaoping Xing, Algebraic Curves in Cryptography Nicholas A. Loehr, Bijective Combinatorics Toufik Mansour, Combinatorics of Set Partitions Toufik Mansour and Matthias Schork, Commutation Relations, Normal Ordering, and Stirling Numbers
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- 10. Pierre-Loïc Méliot Université Paris Sud Orsay, France DISCRETE MATHEMATICS AND ITS APPLICATIONS Representation Theory of Symmetric Groups
- 11. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20170223 International Standard Book Number-13: 978-1-4987-1912-4 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Méliot, Pierre-Loïc, 1985 Title: Representation theory of symmetric groups / Pierre-Loïc Méliot. Description: Boca Raton : CRC Press, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2016050353 | ISBN 9781498719124 Subjects: LCSH: Symmetry groups. | Representations of groups. Classification: LCC QD462.6.S94 M45 2017 | DDC 512/.22--dc23 LC record available at https://lccn.loc.gov/2016050353 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 12. Contents Preface xi I Symmetric groups and symmetric functions 1 1 Representations of finite groups and semisimple algebras 3 1.1 Finite groups and their representations . . . . . . . . . . . . . . . 3 1.2 Characters and constructions on representations . . . . . . . . . 13 1.3 The non-commutative Fourier transform . . . . . . . . . . . . . . 18 1.4 Semisimple algebras and modules . . . . . . . . . . . . . . . . . . 27 1.5 The double commutant theory . . . . . . . . . . . . . . . . . . . . 40 2 Symmetric functions and the Frobenius–Schur isomorphism 49 2.1 Conjugacy classes of the symmetric groups . . . . . . . . . . . . . 50 2.2 The five bases of the algebra of symmetric functions . . . . . . . 54 2.3 The structure of graded self-adjoint Hopf algebra . . . . . . . . . 69 2.4 The Frobenius–Schur isomorphism . . . . . . . . . . . . . . . . . . 78 2.5 The Schur–Weyl duality . . . . . . . . . . . . . . . . . . . . . . . . . 87 3 Combinatorics of partitions and tableaux 99 3.1 Pieri rules and Murnaghan–Nakayama formula . . . . . . . . . . 99 3.2 The Robinson–Schensted–Knuth algorithm . . . . . . . . . . . . . 108 3.3 Construction of the irreducible representations . . . . . . . . . . 131 3.4 The hook-length formula . . . . . . . . . . . . . . . . . . . . . . . . 140 II Hecke algebras and their representations 147 4 Hecke algebras and the Brauer–Cartan theory 149 4.1 Coxeter presentation of symmetric groups . . . . . . . . . . . . . 151 4.2 Representation theory of algebras . . . . . . . . . . . . . . . . . . 161 4.3 Brauer–Cartan deformation theory . . . . . . . . . . . . . . . . . . 173 4.4 Structure of generic and specialized Hecke algebras . . . . . . . 183 4.5 Polynomial construction of the q-Specht modules . . . . . . . . . 207 5 Characters and dualities for Hecke algebras 217 5.1 Quantum groups and their Hopf algebra structure . . . . . . . . 218 5.2 Representation theory of the quantum groups . . . . . . . . . . . 230 5.3 Jimbo–Schur–Weyl duality . . . . . . . . . . . . . . . . . . . . . . . 252 vii
- 13. viii Contents 5.4 Iwahori–Hecke duality . . . . . . . . . . . . . . . . . . . . . . . . . 263 5.5 Hall–Littlewood polynomials and characters of Hecke algebras 272 6 Representations of the Hecke algebras specialized at q = 0 287 6.1 Non-commutative symmetric functions . . . . . . . . . . . . . . . 289 6.2 Quasi-symmetric functions . . . . . . . . . . . . . . . . . . . . . . . 299 6.3 The Hecke–Frobenius–Schur isomorphisms . . . . . . . . . . . . . 306 III Observables of partitions 325 7 The Ivanov–Kerov algebra of observables 327 7.1 The algebra of partial permutations . . . . . . . . . . . . . . . . . 328 7.2 Coordinates of Young diagrams and their moments . . . . . . . . 339 7.3 Change of basis in the algebra of observables . . . . . . . . . . . 347 7.4 Observables and topology of Young diagrams . . . . . . . . . . . 354 8 The Jucys–Murphy elements 375 8.1 The Gelfand–Tsetlin subalgebra of the symmetric group algebra 375 8.2 Jucys–Murphy elements acting on the Gelfand–Tsetlin basis . . 387 8.3 Observables as symmetric functions of the contents . . . . . . . 396 9 Symmetric groups and free probability 401 9.1 Introduction to free probability . . . . . . . . . . . . . . . . . . . . 402 9.2 Free cumulants of Young diagrams . . . . . . . . . . . . . . . . . . 418 9.3 Transition measures and Jucys–Murphy elements . . . . . . . . . 426 9.4 The algebra of admissible set partitions . . . . . . . . . . . . . . . 431 10 The Stanley–Féray formula and Kerov polynomials 451 10.1 New observables of Young diagrams . . . . . . . . . . . . . . . . . 451 10.2 The Stanley–Féray formula for characters of symmetric groups 464 10.3 Combinatorics of the Kerov polynomials . . . . . . . . . . . . . . 479 IV Models of random Young diagrams 499 11 Representations of the infinite symmetric group 501 11.1 Harmonic analysis on the Young graph and extremal characters 502 11.2 The bi-infinite symmetric group and the Olshanski semigroup . 511 11.3 Classification of the admissible representations . . . . . . . . . . 527 11.4 Spherical representations and the GNS construction . . . . . . . 538 12 Asymptotics of central measures 547 12.1 Free quasi-symmetric functions . . . . . . . . . . . . . . . . . . . . 548 12.2 Combinatorics of central measures . . . . . . . . . . . . . . . . . . 562 12.3 Gaussian behavior of the observables . . . . . . . . . . . . . . . . 576
- 14. Contents ix 13 Asymptotics of Plancherel and Schur–Weyl measures 595 13.1 The Plancherel and Schur–Weyl models . . . . . . . . . . . . . . . 596 13.2 Limit shapes of large random Young diagrams . . . . . . . . . . . 602 13.3 Kerov’s central limit theorem for characters . . . . . . . . . . . . 614 Appendix 629 Appendix A Representation theory of semisimple Lie algebras 631 A.1 Nilpotent, solvable and semisimple algebras . . . . . . . . . . . . 631 A.2 Root system of a semisimple complex algebra . . . . . . . . . . . 635 A.3 The highest weight theory . . . . . . . . . . . . . . . . . . . . . . . 641 References 649 Index 661
- 16. Preface The objective of this book is to propose a modern introduction to the representa- tion theory of the symmetric groups. There is now a large literature on the gen- eral representation theory of finite groups, see for instance the classical Linear Representations of Finite Groups by J.-P . Serre ([Ser77]); and among this litera- ture, a few books are concentrated on the case of symmetric groups, for example The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions by B. Sagan (see [Sag01]). The point of view and interest of the present book is the following: we shall show that most of the calculations on symmetric groups can be performed, or at least eased by using some appropriate algebras of functions. It is well known since the works of Frobenius and Schur that the alge- bra of symmetric functions encodes most of the theory of characters of symmetric groups. In this book, we shall use the algebra of symmetric functions as the start- ing point of the representation theory of symmetric groups, and then go forward by introducing other interesting algebras, such as: • the algebra of observables of partitions, originally called “polynomial functions on Young diagrams,” and whose construction is due to Kerov and Olshanski. • the Hopf algebras of non-commutative symmetric functions, quasi-symmetric functions and free quasi-symmetric functions, which contain and generalize the algebra of symmetric functions. This algebraic approach to the representation theory of symmetric groups can be opposed to a more traditional approach which is of combinatorial nature, and which gives a large role to the famous Young tableaux. The approach with algebras of functions has several advantages: 1. First, if one tries to replace the symmetric group by finite-dimensional alge- bras related to it (the so-called partition algebras, or the Hecke algebras), then one can still use the algebra of symmetric functions to treat the charac- ter theory of these algebras, and in this setting, most of the results related to the symmetric groups have direct analogues. In this book, we shall treat the case of Hecke algebras, which is a good example of this kind of extension of the theory of symmetric groups (the case of partition algebras is treated for instance in a recent book by Ceccherini-Silberstein, Scarabotti and Tolli, see [CSST10]). 2. On the other hand, the algebraic approach leads to a new formula for the irreducible characters of the symmetric groups, due to Stanley and Féray. The xi
- 17. xii Preface combinatorics underlying this formula are related to several interesting topics, such as free probability theory, or the theory of Riemann surfaces and maps drawn on them. 3. Finally, the approach with algebras is adequate to deal with asymptotic rep- resentation theory, that is to say representations of symmetric groups S(n) with n large, going to infinity. In this setting, a natural question is: what are the typical properties of a representation of S(n) with n large, and in partic- ular what is the decomposition of such a large representation in irreducible components? Since the irreducible representations of S(n) are labeled by in- teger partitions of size n, this question leads to the study of certain models of random partitions, in particular the so-called Plancherel measures. There, the algebra of observables of partitions will prove a very powerful tool. Besides, our approach enables us to present in a book the theory of combinato- rial Hopf algebras, which is nowadays a quite active field of research in algebraic combinatorics. Let us now detail more precisely the content of the book, which is split into four parts: Â Part I: Symmetric groups and symmetric functions. The first part of the book is devoted to a presentation of the classical theory of representations of symmetric groups, due mainly to Frobenius, Schur and Young. In Chapter 1, we explain the representation theory of finite groups and finite- dimensional semisimple algebras, thereby bringing most of the prerequisites to the reading of the book. One thing that we shall try to do in each chapter is to ob- tain a big “black box theorem,” which summarizes most of the results and allows one to recover at once the remainder of the theory. For the general theory of lin- ear representations of finite groups, one such summarizing statement is the fact that the non-commutative Fourier transform of finite groups is an isomorphism of algebras, of Hilbert spaces and of bimodules (Theorem 1.14). An analogous result holds for finite-dimensional semisimple algebras, the language of algebras and modules being a bit more flexible than the language of groups and represen- tations. In Chapter 2, we introduce the Hopf algebra of symmetric functions Sym, and we show that the Schur functions correspond to the irreducible representa- tions of the symmetric groups: thus, Sym is isomorphic to the Grothendieck ring formed by these representations (Theorem 2.31). This theorem due to Frobenius and Schur can be used as a starting point to the combinatorics of representations, which are developed in Chapter 3 and rely on Young tableaux, that is numberings of Young diagrams of integer partitions. Two other building blocks of this deep combinatorial theory are the Schur–Weyl duality (Section 2.5), which relates the representations of S(n) to the representations of the general linear groups GL(N);
- 18. Preface xiii and the Robinson–Schensted–Knuth algorithm (cf. Section 3.2), which connects the Young tableaux to words or permutations. These two tools will have a perva- sive use throughout the book. Â Part II: Hecke algebras and their representations. In the second part, we explain how one can extend the theory of symmetric groups to other related combinatorial algebras, namely, the so-called Iwahori–Hecke al- gebras. These algebras are continuous deformations Hz(n) of the group algebras CS(n), the parameter z being allowed to take any value in C; one recovers CS(n) when z = 1. In Chapter 4, we show that for almost any value of z, Hz(n) is isomor- phic to CS(n) and has the same representation theory: its irreducible modules Sλ z are again labeled by integer partitions of size n, and they have the same dimension as the irreducible representations Sλ of S(n) (Theorem 4.67). This chapter can be considered as an introduction to modular representation theory that is focused on a specific example. In Chapter 5, we compute the characters of the Hecke algebras in the generic case, by using an extension of Schur–Weyl duality, in which sym- metric groups are replaced by Hecke algebras and linear groups are replaced by quantum groups. We obtain a formula that generalizes the Frobenius–Schur for- mula and involves the Hall–Littlewood symmetric functions (see Theorem 5.49). In Chapter 6, we consider the case z = 0, which is not generic and does not yield a semisimple algebra. In this setting, one can still use combinatorial Hopf algebras to describe the representations of H0(n) (see Theorem 6.18): the algebra of non- commutative symmetric functions NCSym, and the algebra of quasi-symmetric functions QSym, which are in duality. Thus, the extension of the representation theory of symmetric groups to the case of Hecke algebras leads quite naturally to an extension of the theory of symmetric functions to more general functions, which will also appear later in the book (Chapters 10 and 12). Â Part III: Observables of partitions. The third part of the book is devoted to what is now known as the dual combina- torics of the characters of the symmetric groups. In the first part of the book, the characters of the symmetric groups are introduced as functions chλ : S(n) → C or Y(n) → C that are labeled by integer partitions λ of size n, and that can be computed with the help of the Frobenius–Schur formula: chλ (µ) = sλ pµ , where 〈·| ·〉 is the Hall scalar product on the algebra of symmetric functions Sym. However, one can also consider the quantity chλ (µ) as a function of λ labeled by the conjugacy class µ ∈ Y(n). This point of view leads one to consider functions of irreducible representations of symmetric groups, and to introduce an algebra O formed by these functions, which we call the algebra of observables of partitions. Our Chapter 7 presents this algebra and several bases of it, and it explains how the character chλ of the symmetric groups is related to the geometry of the Young diagram of the integer partition λ (see in particular Theorems 7.13 and 7.25). In
- 19. xiv Preface Chapters 8 and 9, we introduce other observables of partitions, related to the so- called Jucys–Murphy elements or to the theory of free probability. In particular, we present an important algebraic basis (Rk)k≥2 of O , whose elements are called free cumulants, and whose combinatorics are related to constructions on set partitions and to maps on surfaces. Chapter 10 explores the interactions between the basis of free cumulants (Rk)k≥2, and the basis of renormalized character values (Σk)k≥1 in O . This study relies on a new formula for the characters of the symmetric groups (Theorem 10.11): chλ (µ) = dimSλ |λ|(|λ| − 1)···(|λ| − |µ| + 1) X ρµ=στ "(τ) Nσ,τ (λ), where the sum runs over factorizations of a permutation ρµ with cycle type µ, and where the quantities Nσ,τ (λ) count certain numberings of the cells of the Young diagram λ. Thus, if instead of Sym one uses the combinatorial algebra O as the starting point of the representation theory of the symmetric groups, then one gets another totally different formula for the irreducible characters, though to be precise the Stanley–Féray formula sits in a larger algebra Q ⊃ O . A care- ful analysis of this formula leads to an explicit change of basis formula between the symbols Rk and the symbols Σk; see Theorem 10.20, which explains how to compute the coefficients of the Kerov polynomials. Â Part IV: Models of random Young diagrams. In the last part of the book, we use the results of the previous chapters in order to describe the properties of the representations of large symmetric groups. In Chapter 11, we start with a classification of the extremal characters of the infinite symmetric groups S(∞) (Theorem 11.31). They play with respect to S(∞) a role similar to the irreducible characters of the finite symmetric groups S(n), and they allow one to consider coherent families (τn)n∈N of representations or more generally of traces of these finite groups. The classification involves an infinite- dimensional convex compact space known as the Thoma simplex. For any param- eter t ∈ T in this simplex, one can consider traces τt,n on the symmetric groups S(n), whose decompositions in irreducible characters yield probability measures Pt,n on the sets Y(n) of integer partitions of size n. Thus, the representation the- ory of S(∞) leads one to study random models of partitions, and this study is performed in Chapters 12 and 13. In Chapter 12, we show that every family of measures (Pt,n)n∈N (the so-called central measures) satisfies a law of large numbers (Theorem 12.19) and a central limit theorem (Theorem 12.30). To this purpose, we introduce a new combinato- rial Hopf algebra FQSym which extends both NCSym and QSym; and a method of joint cumulants of random variables that mixes well with the theory of observ- ables of partitions. In Chapter 13, we study the particular case of Plancherel and Schur–Weyl measures, which have degenerate asymptotics in comparison to the other central measures, and which on the other hand allow one to solve the prob-
- 20. Preface xv lem of the longest increasing subsequences in uniform random permutations or uniform random words (Theorem 13.10). The target audience of this book consists mainly of graduate students and re- searchers. We tried to make the presentation as self-contained as possible, but there remain inevitably certain prerequisites to the reading. Thus, the reader is supposed to have a good familiarity with the basics of algebra (algebraic struc- tures and related constructions) and of combinatorics (counting arguments, bijec- tions); in the last part of the book, we shall also use arguments from probability theory. One prerequisite that helps understanding certain results and that we did not take for granted is the theory of representations of classical Lie algebras; there- fore, Appendix Appendix A is devoted to a short presentation (without proof) of this theory. To be honest, there may be some inconsistencies in the prerequisites that we suppose: for instance, we start the book by recalling what is a group, but later we freely use the language of equivalence of categories. We hope that the long bibliography given at the end of the book will smooth a bit the peaks in difficulty that the reader might encounter. Regarding the bibliography, each chapter is followed by a section called “Notes and references,” where we explain precisely which sources we used in order to write the book. All the credit is due to the authors that are cited in these spe- cial sections, and we tried sincerely not to forget anyone, and to attribute each result to the right mathematician (this task can sometimes be very difficult to ac- complish). Nonetheless, a few results in this book have proofs that are (to our knowledge) either new, or unpublished until now, or very difficult to find in the literature; this is also explained in the notes and references. We also used these special sections to detail some results that we did not have the courage to treat, but that we still wanted to appear in the book. Â Acknowledgments. This book has been built from the contents of my PhD thesis, and from notes of lecture courses that I taught at the University of Zürich in 2012–2013. I am very thankful to my PhD director Philippe Biane for introducing me to the subject of asymptotic representation theory, which is one of the main topics of this work. During the years of preparation of my PhD thesis, I also benefited from the ex- pertise of Jean-Yves Thibon, Jean-Christophe Novelli, Florent Hivert and Alain Lascoux; they introduced me to the theory of combinatorial Hopf algebras, and they showed me how to use them in order to solve many difficult computations. I am much indebted to my colleague Valentin Féray, who explained to me several points of the theory of Kerov polynomials which he developed with Piotr Sniady and Maciej Dołȩga; the discussions that we have are always enlightening. I am grateful to Reda Chhaibi for his explanations on the weight theory and the
- 21. xvi Preface Littelmann path theory of Lie groups and algebras, and for his comments on an early version of the manuscript. Many thanks are also due to Ashkan Nikeghbali, who has a profound influence on the mathematics that I am doing, and invited me numerous times to Zürich. I thank Miklós Bóna for proposing that I write this book, and Bob Ross and José Soto at CRC Press for their assistance and their patience with respect to the numerous small delays that the writing of such a long book caused. I am also very grateful to Karen Simon for supervising the many necessary corrections. Finally, my greatest thanks go to my family, and especially my fiancee Véronique who is a constant support and source of inspiration. Pierre-Loïc Méliot
- 22. Part I Symmetric groups and symmetric functions
- 24. 1 Representations of finite groups and semisimple algebras In this first chapter, we present the general representation theory of finite groups. After an exposition of Maschke’s theorem of complete reducibility of representa- tions (Section 1.1) and of Schur’s lemma of orthogonality of characters (Section 1.2), we construct the non-commutative Fourier transform (Section 1.3), which provides a decomposition of the complex group algebra CG in blocks of endomor- phism rings of the irreducible representations of G. It implies that any function f : G → C can be expanded uniquely as a linear combination of the matrix co- efficients of the irreducible representations of G (Proposition 1.15). This can be seen as a motivation for the study of representations of groups, and on the other hand, the Fourier isomorphism CG → M λ∈b G End(Vλ ) can be generalized to the case of complex semisimple algebras. This language and theory of algebras and modules is in many situations more flexible than the lan- guage of groups and representations, and we devote Section 1.4 to the extension of the theory of representations to this setting. In Section 1.5, this extension allows us to detail the double commutant theory, of which the Frobenius–Schur formula for characters of symmetric groups (see Chapter 2) will be an instance. In the second part of the book, we shall explain the representation theory of some com- binatorial algebras that are deformations of the symmetric group algebra CS(n); there, the knowledge of the representation theory of semisimple algebras will also prove necessary. 1.1 Finite groups and their representations  Finite groups. We assume the reader to be familiar with the notions of groups, rings, fields, vector spaces and algebras. Thus, recall that a group is a set G endowed with an operation ·G : G × G → G (the product of the group), such that 3
- 25. 4 Representations of finite groups and semisimple algebras (G1) ·G is associative and admits a neutral element: ∀g,h, i ∈ G, (g ·G h) ·G i = g ·G (h ·G i); ∃e ∈ G, ∀g ∈ G, g ·G e = e ·G g = g. The neutral element e is then unique. (G2) every element of G has a (unique) inverse for the product: ∀g ∈ G, ∃h ∈ G, g ·G h = h ·G g = e. We shall usually omit the notation ·G, and just denote g·G h = gh. Also, the inverse of g ∈ G will be denoted g−1 , and the neutral element will be denoted indiffer- ently eG, e or 1. A group will be called finite if as a set it has finite cardinality. We then write |G| = card G = number of elements of |G|. Example. Let p be a prime number, and denote Z/pZ = {[1],[2],...,[p]} the set of classes of integers modulo p, which is a ring (quotient of the ring of integers Z). Endowed with the product of classes [a] × [b] = [ab], the set (Z/pZ)∗ = {[1],[2],...,[p −1]} is a finite group of cardinality p −1, with neutral element e = [1]. It is commutative, which means that for any g,h ∈ G, gh = hg. Example. If S is a set, denote S(S) the set of maps σ : S → S that are bijective. This is a group with respect to the operation of composition of maps; the neutral element is the identity idS : s ∈ S 7→ s, and the inverse of a bijection σ ∈ S(S) is the inverse function σ−1 with t = σ(s) if and only if s = σ−1 (t). This book is devoted to the study of the groups S(n) = S([ [1, n] ]), where [ [1, n] ] is the set of integers {1,2,3,..., n − 1, n} between 1 and n. We shall say a bit more about them in a moment. A morphism between two groups G and H is a map φ : G → H compatible with the products of G and H, i.e., such that φ(g1 g2) = φ(g1)φ(g2) for all g1, g2 ∈ G. One speaks of isomorphism of groups if φ is bijective; then, the inverse map φ−1 is also a morphism of groups. On the other hand, a subgroup H of a group G is a subset of G stable by the operations of product and inverse: ∀h1,h2 ∈ H, h1h2 ∈ H ; ∀h ∈ H, h−1 ∈ H. Then, H is a group for the restriction of the product map from G × G to H × H. In the following, we shall say that a group H can be seen as a subgroup of G if there is an injective morphism of groups φ : H → G, which thus identifies H with a subgroup of G.
- 26. Finite groups and their representations 5 Example. Let k be a field, and V be a k-vector space. Then GL(V), the set of bijective linear maps φ : V → V, is a group for the operation of composition of maps. Similarly, if k = C and V is a complex vector space endowed with a scalar product, then the set U(V) of linear isometries of V is a group for the composition of maps, and it is a subgroup of GL(V). Going to the matrix point of view, the following sets of matrices are also groups for the product of matrices, the neutral element being the identity matrix In = diag(1,1,...,1): GL(n, k) = {M ∈ M(n, k) | det(M) 6= 0}; SL(n, k) = {M ∈ M(n, k) | det(M) = 1}; U(n,C) = {M ∈ M(n,C) | M∗ M = M M∗ = In}. If V is a complex vector space of dimension n, then the groups of matrices GL(n,C), SL(n,C) and U(n,C) are isomorphic respectively to GL(V), SL(V) and U(V), the isomorphism being the map which sends a linear map to its matrix in a (unitary) basis. On the other hand, for GL(n, k) to be finite, we need k to be finite, and then, if q = card k, one has cardGL(n, k) = (qn − 1)(qn − q)(qn − q2 )...(qn − qn−1 ). Indeed, this is the number of distinct bases of kn to which an arbitrary basis (e1,..., en) can be sent by an element of GL(n, k) = GL(kn ). Â Symmetric groups. The symmetric group of order n is the group of bijections S(n) = S([ [1, n] ]). It is a finite group with cardinality |S(n)| = n! = 1 × 2 × 3 × ··· × n = n Y i=1 i. Indeed, to choose a bijection σ between elements of [ [1, n] ], one has: • n possibilities for the image σ(1) of 1 (all the integers between 1 and n); • n−1 possibilities for the image σ(2) of 2 (all the integers but the one already chosen for σ(1)); • in general, assuming the images σ(1),...,σ(k) already chosen, n − k possi- bilities for the image σ(k + 1) of k + 1. Multiplying these choices yields |S(n)| = n!. The elements of S(n) are called per- mutations of size n, and we shall denote σ = σ(1)σ(2)...σ(n) a permutation given by the list of its values. So for instance, 4132 is the permutation in S(4) that sends 1 to 4, 2 to 1, 3 to 3 and 4 to 2. Let us now list some easy and well-known properties of these symmetric groups:
- 27. 6 Representations of finite groups and semisimple algebras 1. If n ≤ N, then S(n) can be seen naturally as a subgroup of S(N). Indeed, a bijection σ between the n first integers can be extended in a bijection e σ between the N first integers by setting: e σ(k) = ¨ σ(k) if k ≤ n, k if n < k ≤ N. In the sequel, we shall use these natural imbeddings S(n) ,→ S(N) constantly, and unless the distinction is needed, we shall keep the same notation for a bijection σ ∈ S(n) and its extension to a larger symmetric group S(N). 2. For n ≥ 3, S(n) is a non-commutative group, which means that one can find g and h such that gh 6= hg. Indeed, using the previous property, it suffices to prove the case n = 3, and in this case, if σ = 321 and τ = 213, then their composition products are στ = 231 ; τσ = 312 and they are different. 3. Any finite group can be seen as a subgroup of a finite symmetric group. Indeed, given a finite group G, consider the map φ : G → S(G) g 7→ (φ(g) : h 7→ gh). This is a morphism of groups, which is injective since g can be recovered from the map φ(g) by the formula g = φ(g)(e). On the other hand, given two finite sets A and B with the same cardinality and a bijection ψ : A → B, there is an isomorphism between the groups S(A) and S(B), namely, Ψ : S(A) → S(B) σ 7→ ψ ◦ σ ◦ ψ−1 . Thus, if n = card G, then S(G) and S(n) are isomorphic, so G can be seen as a subgroup of S(n). A more crucial property of permutations deserves the following proposition. Call cycle of length k and support (a1, a2,..., ak) the permutation that sends a1 to a2, a2 to a3, a3 to a4, etc., and ak to a1; and that leaves invariant all the other elements of [ [1, n] ]. For instance, the cycle (1,4,2) in S(4) sends 1 to 4, 4 to 2, 2 to 1, and the remaining element 3 to itself; thus, (1,4,2) = 4132. If k ≥ 2, then a cycle of length k and support (a1,..., ak) is uniquely determined by the sequence (a1, a2,..., ak), up to a cyclic permutation of this sequence: (a1, a2,..., ak) = (a2, a3,..., ak, a1) = (a3,..., ak, a1, a2) = ··· On the other hand, a cycle of length 1 is just the identity permutation, and can be seen as a way to design a particular fixed point (later, we shall make this idea of marked fixed point more rigorous with the notion of partial permutation).
- 28. Finite groups and their representations 7 Proposition 1.1. Any permutation σ ∈ S(n) can be written as a product of cycles with disjoint supports, the sum of the lengths of these cycles being equal to n. This decomposition σ = c1 ◦ c2 ◦ ··· ◦ cr is unique up to permutation of the cycles c1,..., cr . Proof. In a finite group, every element g has for inverse a positive power of itself gk≥1 . As a consequence, the permutation σ ∈ S(n) being fixed, the relation on [ [1, n] ] defined by i ∼σ j ⇐⇒ ∃k ≥ 0, j = σk (i), which is clearly reflexive and transitive, is also symmetric, so it is an equivalence relation. Call orbit of σ a class for the equivalence relation ∼σ on [ [1, n] ]; then, the orbit of i, if it has length k, is {i,σ(i),σ2 (i),...,σk−1 (i)}, and the restriction of σ to this orbit is the cycle c = (i,σ(i),σ2 (i),...,σk−1 (i)). The decomposition of σ in disjoint cycles is then obtained by choosing one representative for each orbit, and the unicity comes from the fact that if σ = c1 ◦ c2 ◦ ··· ◦ cr is a product of cycles with disjoint supports, then these supports are orbits of σ, with the order of elements for each cycle entirely determined by the action of σ on each support. Example. Consider the permutation σ = 874312659 in S(9). Its orbits are {1,5,8}, {2,6,7}, {3,4} and {9}, and the cycle decomposition of σ is σ = (1,8,5)(2,7,6)(3,4)(9). Thus, we get two different writings for a given permutation σ ∈ S(n): the notation in line σ = σ(1)σ(2)...σ(n), and the cycle decomposition σ = (a1,..., ar )(b1,..., bs)···(z1,...,zt ). In the cycle decomposition, it will sometimes be convenient to omit the cycles of length 1, since they correspond to the identity. This is in particular the case if σ is itself a single cycle. We say that σ is a transposition if it is a cycle of length 2; then, it writes as σ = (i, j) = (j, i) and it exchanges i and j. Â Representations of groups. If V is a complex vector space, we denote as before GL(V) the group of complex linear isomorphisms u : V → V. If V is finite-dimensional and if (e1,..., en) is a fixed linear basis of V, we denote (ui j)1≤i,j≤n the matrix of the linear map u in this basis, which means that u(ej) = n X i=1 ui j ei.
- 29. 8 Representations of finite groups and semisimple algebras Then, the map ψ : u ∈ GL(V) 7→ (ui j)1≤i,j≤n ∈ GL(n,C) is an isomorphism of groups. Definition 1.2. A (complex, linear) representation of a group G is given by a complex vector space V, and a morphism of groups ρ : G → GL(V). We shall always assume the space V to be finite-dimensional, and we shall denote (ρi j(g))1≤i,j≤dim V the matrix of ρ(g) in a fixed basis of the representation. For every g ∈ G, ρ(g) is a linear isomorphism of V, and we can make it act on vectors v ∈ V. Thus, we shall frequently manipulate vectors (ρ(g))(v) ∈ V with g ∈ G and v ∈ V, and if the representation (V,ρ) is fixed, we shall abbreviate (ρ(g))(v) = g · v. Then, to describe a representation of G amounts to giving a formula for g ·v, with the condition that g · (h · v) = (gh) · v for any g,h ∈ G and any v ∈ V. Notice then that for any v ∈ V, 1 · v = v if 1 denotes the neutral element of G. Example. For any group G, one has the so-called trivial representation of G on V = C, given by g · v = v for any g ∈ G and any v ∈ V. Example. Fix a positive integer n, and consider the permutation representation of S(n) on Cn , given by σ · (x1,..., xn) = (xσ−1(1),..., xσ−1(n)). This is indeed a representation, since σ · (τ · (x1,..., xn)) = σ · (xτ−1(1),..., xτ−1(n)) = (xτ−1σ−1(1),..., xτ−1σ−1(n)) = (x(στ)−1(1),..., x(στ)−1(n)) = (στ) · (x1,..., xn). The matrix of ρ(σ) in the canonical basis of Cn is the permutation matrix (δi,σ(j))1≤i,j≤n, where δa,b denotes the Dirac function, equal to 1 if a = b and to 0 otherwise (this notation will be used throughout the whole book). Example. Let G be a finite group. We denote CG the vector space of functions from G to C, and we identify a function f with the formal linear sum f = X g∈G f (g) g. So for instance, if G = S(3) = {123,132,213,231,312,321}, then 2(123) − (213) + (1 + i)(321)
- 30. Finite groups and their representations 9 represents the function which sends 123 to 2, 213 to −1, 321 to 1 + i, and the other permutations in S(3) to 0. With these notations, a basis of CG is G, an element g ∈ G being identified with the Dirac function δg. The (left) regular representation of G is the representation with space V = CG, and with g · f (h) = f (g−1 h), which writes more easily as g · ‚ X h∈G f (h) h Œ = X h∈G f (h) gh. The morphism underlying this regular representation is the composition of the morphism G → S(G) described in the previous paragraph, and of the permutation representation S(G) → GL(CG).  Irreducible representations and Maschke’s theorem. A subrepresentation of a representation (V,ρ) is a vector subspace W ⊂ V that is stable by the action of G, which means that ∀g ∈ G, ∀v ∈ W, g · v ∈ W. Then, W is a representation of G for the new morphism ρ|W (g) = (ρ(g))|W . A representation (V,ρ) of G is said to be irreducible if it has positive dimension and if there is no stable subspace (subrepresentation) W ⊂ V with W 6= {0} and W 6= V. As we shall see at the end of this paragraph, any representation of a finite group can be split into smaller irreducible representations. Let us first detail the notions of morphism of representations and of direct sum of representations. If V1 and V2 are two representations of G, then their direct sum is the representa- tion of G with underlying vector space V1 ⊕ V2, and with g · (v1 + v2) = g · v1 + g · v2 for any g ∈ G and any (v1, v2) ∈ V1 × V2. On the other hand, given again two representations V1 and V2, a morphism of representations between V1 and V2 is a linear map φ : V1 → V2 such that φ(g · v) = g · φ(v) for any g ∈ G and any v ∈ V. Thus, for any element of the group, the following diagram of linear maps is commutative: V1 ρ1(g) − −−− → V1 φ y yφ V2 − −−− → ρ2(g) V2
- 31. 10 Representations of finite groups and semisimple algebras Example. Consider the permutation representation of S(n) on Cn . It admits as stable subspaces V1 = {(x1,..., xn) ∈ Cn | x1 + ··· + xn = 0}; V2 = C(1,1,...,1), and Cn is the direct sum of these two representations: Cn = V1 ⊕ V2. It is clear for dimension reasons that V2 is irreducible; we shall see later that V1 is also irreducible. Notice that V2 is isomorphic to the trivial representation of S(n), the isomorphism being given by λ(1,1,...,1) 7→ λ. Theorem 1.3 (Maschke). Let G be a finite group and V be a (finite-dimensional) representation of G. 1. There exists a decomposition of V as a direct sum of irreducible representations of G: V = r M i=1 Vi, with each Vi irreducible representation of G. 2. Fix an irreducible representation I of G. The number of components Vi of V that are isomorphic to I is independent of the decomposition of V in irreducible repre- sentations. Moreover, the regular representation CG of G has dim I components isomorphic to I. Before we prove it, let us restate in a clearer way the consequences of Theorem 1.3. There exists a decomposition of the regular representation CG = M λ∈b G dλVλ , where b G is a finite set; each Vλ is an irreducible representation appearing with multiplicity dλ = dim Vλ ; and two representations Vλ and Vµ with λ 6= µ are non- isomorphic. Then, every other representation V of G writes up to an isomorphism of representations as V = M λ∈b G mλVλ , with the multiplicities mλ ∈ N uniquely determined by V. The proof of Theorem 1.3 relies on the two following lemmas: Lemma 1.4. Let (V,ρ) be a representation of a finite group G. There exists a scalar product 〈·| ·〉 on V such that ρ(g) ∈ U(V) for any g ∈ G: ∀v1, v2 ∈ V, 〈g · v1 | g · v2〉 = 〈v1 | v2〉.
- 32. Finite groups and their representations 11 Remark. In this book, every instance of a Hermitian scalar product 〈·| ·〉 will be antilinear in the first variable, and linear in the second variable. Thus, if v and w are in V and a, b ∈ C, then 〈av | bw〉 = ab 〈v | w〉. Proof. We start with an arbitrary scalar product (·|·) on V, and consider the new scalar product 〈v1 | v2〉 = X g∈G (g · v1|g · v2). Then, 〈·| ·〉 is obviously again a scalar product, and 〈h · v1 | h · v2〉 = X g∈G (gh · v1|gh · v2) = X g∈G (g · v1|g · v2) = 〈v1 | v2〉 for any h ∈ G. Lemma 1.5 (Schur). Given two representations V and W of a finite group G, denote HomG(V,W) the vector space of morphisms of representations between V and W. If V and W are irreducible, then dimHomG(V,W) = ¨ 1 if V and W are isomorphic; 0 otherwise. On the other hand, for any representation V of a finite group G, there is an isomor- phism of vector spaces between V and HomG(CG, V). Proof. For any morphism of representations φ : V → W, the kernel and the image of φ are subrepresentations respectively of V and of W. Fix then an irreducible representation V, and a morphism of representations φ : V → V. For any λ ∈ C, φ − λidV is also an endomorphism of representations. Take λ among the non- empty set of eigenvalues of φ: then, Ker(φ − λidV ) is a subrepresentation of V, so it is equal to V and φ = λidV . Consider then another irreducible representation W of G. If V and W are isomorphic by ψ : V → W, then for any morphism of representations φ : V → W, ψ−1 ◦φ ∈ HomG(V, V), so it is a multiple of idV , and φ = λψ for some scalar λ. So, dimHomG(V,W) = 1 if V and W are isomorphic. If V and W are not isomorphic, then given a morphism of representations φ : V → W, either its kernel is non- zero, or its image is not equal to W. By irreducibility, this implies that either Kerφ = {0} or Imφ = {0}, so φ = 0, and the second case for the computation of dimHomG(V,W) is treated. Finally, consider a representation V of G and a vector v ∈ V. The map φ : CG → V X g∈G f (g) g 7→ X g∈G f (g)(g · v)
- 33. 12 Representations of finite groups and semisimple algebras is a morphism of representations between CG and V, and conversely, given a morphism φ ∈ HomG(CG, V), it is easy to see that it is given by the previous formula for v = φ(e). Thus, we have a natural identification between V and HomG(CG, V). Proof of Theorem 1.3. Let V be a representation of G, endowed with a G-invar- iant scalar product as in Lemma 1.4. If V is not itself irreducible, consider a stable subspace V1 ⊂ V with V1 6= {0} and V1 6= V. The orthogonal V2 = (V1)⊥ is also stable: if v ∈ V2, then for any g ∈ G and any w ∈ V1, 〈g · v | w〉 = v g−1 · w = 0 since g−1 · w ∈ V1 and v ∈ V2, so g · v ∈ V2. Thus, we have the decomposition in stable subspaces V = V1 ⊕ V2, and by induction on the dimension of V, the representation V can be totally split in irreducible representations. For the second part of the theorem, since HomG(·,·) is compatible with direct sums, if V = Lr i=1 Vi is a decomposition in irreducible representations of V, then given another irreducible representation I, by Schur’s lemma, dimHomG(V, I) = number of indices i such that I and Vi are isomorphic. This irreducible representation I is also always a component of CG, with multi- plicity dimHomG(CG, I) = dim I > 0 by the second part of Lemma 1.5. Remark. From the proof of Theorem 1.3, we see that if V = L λ∈b G mλVλ is a decomposition of V into non-isomorphic irreducible representations (with multi- plicities), then mλ = dimHomG(V, Vλ ). Let us reformulate once more the content of Theorem 1.3. We shall always denote b G the set of non-isomorphic irreducible representations λ = (Vλ ,ρλ ) ap- pearing as components of the regular representation CG. By the previous discus- sion, they are also the irreducible components of all the representations of G. Now, consider the set S(G) of classes of isomorphism of representations of G. The operation of direct sum ⊕ makes S(G) into a commutative monoid, with neutral element the class of the null representation {0}. Denote R0(G) the Grothendieck group built from S(G), that is to say, the set of formal differences V W of (classes of isomorphism of) representations of G, with V1 W1 = V2 W2 if and only if V1 ⊕ W2 and V2 ⊕ W1 are isomorphic, and (V1 W1) ⊕ (V2 W2) = (V1 ⊕ V2) (W1 ⊕ W2). We call R0(G) the Grothendieck group of representations of G.
- 34. Characters and constructions on representations 13 Proposition 1.6. For any finite group G, R0(G) = M λ∈b G ZVλ , and the elements of R0(G) with non-negative coefficients correspond to classes of isomorphism of representations of G. The main result of Chapter 2 will be a description of the Grothendieck groups of representations of the symmetric groups S(n). 1.2 Characters and constructions on representations  Characters and Schur’s lemma of orthogonality. From the previous paragraph, we know that a linear representation of a finite group G is entirely determined up to isomorphisms by a finite sequence of non- negative numbers (mλ)λ∈b G. However, these numbers have been described so far as dimensions of spaces of morphisms of representations, and one may ask for a simpler way to compute them in terms of V. The theory of characters yields a convenient tool in this setting. Definition 1.7. The character of a representation (V,ρ) of G is defined by chV (g) = tr(ρ(g)). Thus, if (ρi j(g))i,j is the matrix of ρ(g) in a basis of V, then the character is chV (G) = Pdim V i=1 ρii(g). In many situations, it will be also useful to deal with the normalized character χV of a representation: it is defined by χV (g) = chV (g) chV (1) = chV (g) dim V . Example. Consider the regular representation of a finite group G. Its character is chV (g) = X h∈G δh,gh = ¨ |G| if g = 1, 0 otherwise. Notice that for any representation V, chV (g−1 ) = chV (g). Indeed, we can write the matrices of representation in a basis of V that is unitary with respect to a G-invariant scalar product on V. Then, chV (g) = trρ(g) = tr(ρ(g))∗ = tr(ρ(g))−1 = chV (g−1).
- 35. 14 Representations of finite groups and semisimple algebras On the other hand, the characters always have the trace property: ∀g,h ∈ G, chV (gh) = tr(ρ(g)ρ(h)) = tr(ρ(h)ρ(g)) = chV (hg). We view the characters as elements of CG, and we endow this space of functions on G with the scalar product 〈f1 | f2〉G = 1 |G| X g∈G f1(g) f2(g). For any irreducible representation λ ∈ b G, we fix a G-invariant scalar product on Vλ , and we denote (ρλ i j(g))1≤i,j≤dλ the matrix of ρ(g) in a unitary basis of Vλ . We also write chλ for the irreducible character of Vλ ; thus, chλ = Pdλ i=1 ρλ ii, where dλ = dim Vλ . Theorem 1.8 (Schur). For any irreducible representations λ and µ of G, ¬ ρλ i j ρ µ kl ¶ G = 1 dim Vλ δλ,µ δi,k δj,l. As a consequence, chλ chµ G = δλ,µ. Proof. Let u be an arbitrary linear map between Vµ and Vλ . We set φ = 1 card G X g∈G ρλ (g) ◦ u ◦ ρµ (g−1 ). This map is a morphism of representations between Vµ and Vλ . Indeed, for any h ∈ G, φ ◦ ρµ (h) = 1 |G| X g∈G ρλ (g) ◦ u ◦ ρµ (g−1 h) = 1 |G| X k∈G ρλ (hk) ◦ u ◦ ρµ (k−1 ) = ρλ (h) ◦ φ. Since Vµ and Vλ are irreducible representations, the previous map φ is 0 unless λ = µ. In this latter case, if one makes the identification Vλ = Vµ , then φ is a scalar multiple of the identity, and the coefficient of proportionality can be found by taking the trace: φ = λidVλ with λ(dim Vλ ) = trφ = tru. Now, let us write the representations in matrix form. The previous computations become: 1 |G| X g∈G dλ X j=1 dµ X k=1 ρλ i j(g)ujk ρ µ kl (g−1 ) = ¨ δi,l dim Vλ Pdλ j=1 uj j if λ = µ, 0 otherwise.
- 36. Characters and constructions on representations 15 Both sides of the equation are linear forms in the coefficients of u, so their coefficients must be equal. Therefore, for any coefficients i, j ∈ [ [1, dλ] ] and k, l ∈ [ [1, dµ] ], ¬ ρλ i j ρ µ lk ¶ G = 1 |G| X g∈G ρλ i j(g)ρ µ lk (g) = 1 |G| X g∈G ρλ i j(g−1 )ρ µ kl (g) = ¨δi.l δj,k dim Vλ if λ = µ, 0 otherwise. This is the equality stated by the theorem with the roles of k and l exchanged. By taking i = j and k = l, and summing over indices i ∈ [ [1, dλ] ] and k ∈ [ [1, dµ] ], we get the orthogonality relation for characters. Corollary 1.9. A representation V of a finite group G is entirely determined (up to isomorphisms) by its character. Proof. If V = L λ∈b G mλ Vλ , then chV = P λ∈b G mλ chλ . By using the orthogonality of characters, one sees then that the multiplicity mλ of an irreducible represen- tation Vλ in V is given by mλ = chV chλ G . As a by-product, one gets a cri- terion of irreducibility for a representation of G: V is irreducible if and only if chV chV G = 1. Example. Consider the permutation representation of S(n) on Cn , with n ≥ 2. Its character is chCn (σ) = n X i=1 δi,σ(i) = number of fixed points of σ. Let us compute the norm of this character: chCn chCn S(n) = 1 n! X σ∈S(n) ‚ n X i=1 δi,σ(i) Œ2 = 1 n! n X i,j=1 X σ∈S(n) δi,σ(i) δj,σ(j) ! = 1 n! n X i=1 (n − 1)! + X 1≤i6=j≤n (n − 2)! ! = n! + n! n! = 2. Indeed, on the second line, if two indices i and j are fixed, then the number of permutations with σ(i) = i and σ(j) = j is (n − 2)!; and similarly, if one index i is fixed, then the number of permutations with σ(i) = i is (n − 1)!. It follows that Cn is necessarily the direct sum of 2 non-isomorphic irreducible representations; indeed, if V = L λ∈b G mλ Vλ , then chV chV G = P λ∈b G (mλ)2 . By the discussion on page 10, these two components are the trivial representation on C(1,1,...,1), and the representation on {(x1,..., xn) ∈ Cn | x1 + ··· + xn = 0}.
- 37. 16 Representations of finite groups and semisimple algebras  Tensor products, induction and restriction of representations. Because of Corollary 1.9, each statement on representations has an equivalent statement in terms of characters, and from now on we shall try to give both state- ments at each time. For instance, let us present some constructions on represen- tations, and their effect on characters. We already used the fact that the direct sum of representations corresponds to the sum of characters: ch(V⊕W) = chV + chW . The tensor product of two representations V and W of G is the representation with underlying vector space V ⊗ W, and with g · (v ⊗ w) = (g · v) ⊗ (g · w). Since simple tensors v ⊗ w span linearly V ⊗ W, the previous formula entirely defines a representation on V ⊗ W. Proposition 1.10. The character of a tensor product of representations is the prod- uct of the characters: ch(V⊗W) = chV × chW . Proof. Fix a basis (e1,..., em) of V, and a basis (f1,..., fn) of W. A basis of V ⊗W is (ei ⊗ fj)i,j, and with respect to these bases, ρV⊗W (i,j)(k,l) (g) = ρV ik(g)ρW jl (g). Therefore, chV⊗W (g) = X i,j ρV⊗W (i,j)(i,j) (g) = ‚ X i ρV ii (g) Œ X j ρW j j (g) ! = chV (g) chW (g). Consider now two finite groups H ⊂ G. There is a canonical way to transform representations of H into representations of G and conversely. First, if (V,ρV ) is a representation of G, then the restricted representation ResG H (V) is the repre- sentation of H defined by ∀h ∈ H, ρResG H (V) (h) = ρV (h). Thus, it has the same underlying vector space as V, and the action of H is just obtained by restriction of the definition of the action of G. In particular, chResG H (V) = chV |H .
- 38. Characters and constructions on representations 17 The converse operation of induction from H to G is a bit more cumbersome to define without the language of algebras and modules. If (V,ρV ) is a representation of H, denote IndG H (V) the set of functions f : G → V such that ∀h ∈ H, ∀g ∈ G, f (hg) = h · f (g). If one fixes a set of representatives f g1,..., e gr of HG, then a function in IndG H (V) is determined by its values on f g1,..., e gr , so dimIndG H (V) = [G : H](dim V) = |G| |H| dim V. We make G act on IndG H (V) by (g · f )(g0 ) = f (g0 g), and we call IndG H (V) the induced representation of V from H to G. Proposition 1.11. The character of an induced representation IndG H (V) is given by the formula chIndG H (V) (g) = X gj ∈G/H chV (g−1 j ggj), where chV (g) = 0 if g is not in H, and the sum runs over representatives g1,..., gr of the left cosets gH. We postpone the proof of this proposition to the end of Section 1.4, where the framework of algebras and modules will provide a more natural definition of the induction of representations, and a simple explanation of the formula for characters. Â Frobenius’ reciprocity. An important feature of the operations of induction and restriction of representa- tions is their adjointness in the sense of functors on categories. More concretely, one has: Proposition 1.12 (Frobenius). Let H ⊂ G be two finite groups, and V and W be two representations of H and G. One has ¬ chV chResG H (W) ¶ H = ¬ chIndG H (V) chW ¶ G . In particular, if V and W are irreducible, then the multiplicity of V in ResG H (W) is the same as the multiplicity of W in IndG H (V).
- 39. 18 Representations of finite groups and semisimple algebras Proof. We compute: ¬ chIndG H (V) chW ¶ G = 1 |G| X g∈G chIndG H (V) (g)chW (g) = 1 |G| X g∈G X gj ∈G/H chV (g−1 j ggj) chW (g) = 1 |G| X k∈G X gj ∈G/H chV (k) chW (gj kg−1 j ) because when g runs over G, so does k = g−1 j ggj. Now, by the trace property of characters, chW (gj kg−1 j ) = chW (k), so ¬ chIndG H (V) chW ¶ G = 1 |G| X k∈G X gj ∈G/H chV (k) chW (k) = 1 |H| X k∈G chV (k) chW (k). Finally, since chV (k) = 0 if k is not in H, the last sum runs in fact over H, and we get indeed ¬ chV chResG H (W) ¶ H . When we shall deal with representations of symmetric groups (Chapter 2), Frobenius’ reciprocity will translate into a property of self-adjointness for the Hopf algebra of symmetric functions. 1.3 The non-commutative Fourier transform  The Fourier transform and the algebra Cb G. For any group G, Theorem 1.3 ensures that there is an isomorphism of represen- tations CG → M λ∈b G dλVλ , where b G is the finite set of all classes of isomorphism of irreducible representations of G. However, this isomorphism is for the moment an abstract one, and one may ask for a concrete realization of it. This realization will be provided by the so- called non-commutative Fourier transform, whose properties will allow us to restate and summarize most of the previous discussion. The vector space CG can be endowed with a structure of algebra for the con- volution product: (f1 f2)(k) = X gh=k f1(g) f2(h).
- 40. The non-commutative Fourier transform 19 This rule of product is easy to understand if one identifies as before a function f with the formal sum P g∈G f (g) g: f1 f2 = X g∈G f1(g) !‚ X h∈G f2(h) Œ = X g,h∈G f1(g)f2(h) gh = X k∈G X gh=k f1(g) f2(h) ! k. We say that CG is the group algebra of G. On the other hand, we denote Cb G the complex algebra which is the direct sum of all the algebras End(Vλ ): Cb G = M λ∈b G End(Vλ ). It is again convenient to see formal sums of endomorphisms in Cb G as functions on b G. Definition 1.13. The non-commutative Fourier transform b f of a function f ∈ CG is the element of Cb G defined by b f (λ) = X g∈G f (g)ρλ (g). Example. Consider the symmetric group S(3). We already know two non- isomorphic irreducible representations of S(3): V1 = {(x1, x2, x3) ∈ C3 | x1 + x2 + x3} ; V2 = C; the first representation being the (restriction of) the permutation representation, and the second representation being the trivial one. A third irreducible represen- tation is provided by the signature representation (cf. Section 2.1) V3 = C ; ρ(σ) = (σ) = (−1) P ij δσ(i)σ(j) ∈ C× = GL(1,C). It is of dimension 1, hence irreducible, and equal to its character, which is different from the trivial character; so, it is non-isomorphic to V1 and V2. Since cardS(3) = 6 = 22 + 12 + 12 = (dim V1)2 + (dim V2)2 + (dim V3)2 , we thus have a complete set of representatives of Ò S(3). Denoting (ρi j(σ))i,j the 2 × 2 matrix of the representation V1, the Fourier transform of a permutation σ ∈ S(3) can thus be seen as the block-diagonal matrix ρ11(σ) ρ12(σ) ρ21(σ) ρ22(σ) 1 (σ) .
- 41. 20 Representations of finite groups and semisimple algebras  The fundamental isomorphism. The space Cb G is a representation of G for the action g· P λ∈b G uλ = P λ∈b G ρλ (g)uλ . Also, it admits for G-invariant scalar product 〈u| v〉b G = X λ∈b G dλ |G|2 tr((uλ )∗ vλ ), the adjoint of an endomorphism in each space End(Vλ ) being taken with respect to a G-invariant scalar product on Vλ . In the following, we fix for each λ a unitary basis (eλ i )1≤i≤dλ of each space Vλ , and denote (eλ i j)1≤i,j≤dλ the associated basis of End(Vλ ): eλ i j(eλ k ) = δj,k eλ i . Theorem 1.14. The Fourier transform CG → Cb G is an isomorphism of algebras, of representations of G, and of Hilbert spaces. The matrix coefficients of irreducible rep- resentations (ρλ i j)λ∈b G, 1≤i,j≤dλ form an orthogonal basis of CG. If ηλ i j(g) = ρλ ji(g−1 ), then this new orthogonal basis (ηλ i j) is sent by the Fourier transform to Ó ηλ i j = |G| dλ eλ i j. Proof. We saw in Theorem 1.8 that the matrix coefficients of irreducible repre- sentations are orthogonal, and since |G| = dimCG = X λ∈b G (dλ)2 , we have the right number of terms to form a basis of CG. On the other hand, the Fourier transform is indeed compatible with the product on each algebra: Ô f1 f2 = X λ∈b G X k∈G (f1 f2)(k) ρλ (k) = X λ∈b G X g,h∈G f1(g)f2(h) ρλ (gh) = X λ∈b G X g∈G f1(g)ρλ (g) !‚ X h∈G f2(h)ρλ (h) Œ = X λ∈b G Ò f1(λ) Ò f2(λ) = Ò f1 Ò f2. The compatibility with the action of G is trivial. Suppose that b f = 0. Then, for any irreducible representation λ and any indices 1 ≤ i, j ≤ dλ, ¬ f ρλ i j ¶ G = X i,j f (g)ρλ i j(g) = (b f (λ))i j = 0, so f is orthogonal to all the elements of an orthogonal basis, and f = f = 0. It follows that the Fourier transform is injective, and since dimCb G = dimCG, it is an isomorphism of algebras and of representations of G.
- 42. The non-commutative Fourier transform 21 To prove that it is also an isomorphism of Hilbert spaces, it suffices to show that for any g,h ∈ G, δg δh G = ¬ c δg c δh ¶ b G ; indeed, the functions δg form an orthogonal basis of CG. Notice that δg δh G = δg,h |G| = δe,g−1h |G| = chCG (g−1 h) |G|2 . However, ¬ c δg c δh ¶ b G = X λ∈b G dλ |G|2 tr (ρλ (g))∗ ρλ (h) = X λ∈b G dλ |G|2 tr ρλ (g−1 h) = 1 |G|2 X λ∈b G dλ chVλ (g−1 h) = chCG (g−1 h) |G|2 , the last identity coming from the isomorphism CG = L λ∈b G dλVλ . Finally, we compute the Fourier transform of a matrix coefficient of irreducible representations: Ó ηλ i j = X µ∈b G X g∈G ρλ i j(g) ρµ (g) = |G| X µ∈b G X 1≤k,l≤dµ ¬ ρλ i j ρ µ kl ¶ G e µ kl = |G| dλ eλ i j.  Decomposition of functions on groups. An important consequence of Theorem 1.14 is the possibility to expand every function on the group G as a linear combination of matrix coefficients of irre- ducible representations: f (g) = X λ,i,j dλ ¬ ηλ i j f ¶ G ηλ i j(g) = X λ∈b G dλ |G| X 1≤i,j≤dλ ‚ X h∈G f (h)ρλ i j(h) Œ ρλ i j(g) = X λ∈b G dλ |G| X 1≤i,j≤dλ b f (λ) i j (ρλ ji)∗ (g) = X λ∈b G dλ |G| tr ρλ∗ (g) b f (λ) . To get a good intuition of these results, it can be useful to compare this expan- sion of functions with the usual Fourier theory of functions on a circle (or on a multi-dimensional torus). To this purpose, it is convenient to renormalize a bit the algebra structures on CG and Cb G. These modifications will only hold during this paragraph. To avoid any ambiguity, the dual elements (irreducible represen- tations, Fourier transforms) will be denoted in this paragraph with a symbol e · instead of b ·. We renormalize the convolution product on CG by setting (f1 ∗ f2)(k) = 1 |G| X gh=k f1(g) f2(h),
- 43. 22 Representations of finite groups and semisimple algebras and we keep the same Hilbert scalar product on CG as before. We define as before the dual group algebra Ce G = L λ∈e G End(Vλ ), but we change the scalar product of Ce G into 〈u| v〉e G = X λ∈e G dλ tr((uλ )∗ vλ ). We define the Fourier transform of a function f by e f (λ) = 1 |G| X g∈G f (g)ρλ (g). Then, the new Fourier transform f 7→ e f is as before an isomorphism of C-algebras, of G-representations, and of Hilbert spaces. Moreover: Proposition 1.15. For any function f on the group G, f (g) = X λ∈e G dλ tr ρλ∗ (g) e f (λ) ; 〈f | f 〉G = e f e f e G = X λ∈e G dλ tr (e f (λ))∗ e f (λ) . These formulas are exactly the same as those satisfied by the Fourier series of a square-integrable function f on the circle T = R/(2πZ): f (θ) = X k∈Z e f (k)e−ikθ ; 1 2π Z 2π 0 |f (θ)|2 dθ = X k∈Z e f (k) 2 where e f (k) = 1 2π R 2π 0 f (θ)eikθ dθ. The reason for this correspondence is that the formulas of Proposition 1.15 hold in fact for any square-integrable function on a topological compact group, the means 1 |G| X g∈G · being replaced in this theory by integrals Z G · Haar(dg) against the Haar measure. For instance, with the circle T, the set of irreducible representations is labeled by Z, each irreducible representation of label k ∈ Z be- ing one-dimensional and given by ρ(θ) = eikθ . The Haar measure on T = [0,2π] is dθ 2π and Proposition 1.15 gives indeed the Fourier series of harmonic functions. We leave as an exercise to the reader (see also the notes at the end of the chapter) the proof that almost all results proved so far for representations and characters of finite groups extend to topological compact groups, the only difference being that
- 44. The non-commutative Fourier transform 23 the set b G of irreducible representations appearing as components of the regular representation L2 (G,Haar) can now be discrete infinite (this being a consequence of the spectral theory of compact operators). In particular, most of the theory exposed before can be applied without big changes to the classical compact Lie groups SU(n), SO(n), USp(n). The fact that the coefficients of representations yield an expansion of arbitrary functions in orthogonal components can be seen as one of the main motivations for the study of representations of groups. In particular, it enables one to solve evolution problems such as the heat equation on non-commutative groups (in- stead of the basic setting of the real line). In the next paragraph, we shall give an example of this in the case of finite groups. Â Center of the group algebra. In this paragraph, we consider the restriction of the non-commutative Fourier transform to the center of the group algebra CG. Write Z(CG) for the set of functions on G such that f (gh) = f (hg) for any g,h. Lemma 1.16. The following assertions are equivalent: (Z1) The function f belongs to Z(CG). (Z2) The function f commutes with any other function d of CG: f d = d f . (Z3) The function f is a linear combination of conjugacy classes Cg = X g0=h−1 gh g0 . Proof. If f ∈ Z(CG), then f (h−1 gh) = f (hh−1 g) = f (g) for any g,h, so f is constant on conjugacy classes, and this proves the equivalence between (Z1) and (Z3). Then, if f ∈ Z(CG), one has for any other function d f d = X g,h∈G f (g) d(h) gh = X g,h∈G f (h−1 gh) d(h)hh−1 gh = X h,g0∈G d(h) f (g0 )hg0 = d f , so (Z1) ⇒ (Z2). Conversely, if f commutes with any other function, then f (gh) = (f δh−1 )(g) = (δh−1 f )(g) = f (hg). Since CG and Cb G are isomorphic by the Fourier transform, their centers are isomorphic, and the center of an endomorphism algebra End(Vλ ) is the one- dimensional space CidVλ . It will be convenient to identify an element of Z(Cb G) = M λ∈b G CidVλ
- 45. 24 Representations of finite groups and semisimple algebras with a C-valued function on b G, according to the following rule: (k : b G → C) corresponds to X λ k(λ)idVλ . Then, the restriction of the scalar product 〈·| ·〉b G to Z(Cb G) is defined on functions b G → C by 〈k1 | k2〉b G = X λ∈b G dλ |G|2 tr € k1(λ) k2(λ)idVλ Š = X λ∈b G dλ |G| ‹2 k1(λ) k2(λ). In the following, for any function f in the center, we set f ∗ (g) = f (g−1 ). Notice that (chV )∗ = chV for any character of representation. Theorem 1.14 restricted to Z(CG) reads now as: Theorem 1.17. Redefine the Fourier transform of an element f ∈ Z(CG) as the function b f (λ) = X g∈G f (g)χλ (g). Then, the Fourier transform is an isometry between Z(CG) and Z(Cb G) = C b G . An orthonormal basis of Z(CG) consists in the irreducible characters chλ , and the image of chλ∗ by the Fourier transform is the function Ô chλ∗ = |G| dλ δλ. Proof. The redefinition of the Fourier transform on Z(CG) is compatible with the definition used in Theorem 1.14: b f (λ) = tr b f (λ) dλ idλ V = X g∈G f (g)χλ (g) ! idVλ . Hence, the first part is an immediate consequence of Theorem 1.14. Then, we compute Ô chλ∗ (µ) = X g∈G chλ (g)χµ (g) = |G| dµ chλ chµ G = |G| dλ δλ,µ. Corollary 1.18. The number of distinct irreducible representations in b G is the num- ber of conjugacy classes of the group G. Proof. This is the dimension of Z(Cb G), which is isomorphic to Z(CG). Corollary 1.19. Any central function f expands on irreducible characters as f (g) = X λ∈b G (dλ)2 |G| c f ∗(λ)χλ (g).
- 46. The non-commutative Fourier transform 25 Proof. Since irreducible characters form an orthonormal basis of Z(CG), f (g) = X λ∈b G chλ f G chλ (g) = X λ∈Ĝ (dλ)2 χλ f G χλ (g). Then, χλ f G = 1 |G| P g∈G f (g)χλ (g−1 ) = 1 |G| P g∈G f ∗ (g)χλ (g) = c f ∗(λ) |G| . The last result involves the so-called Plancherel measure Pl(λ) = (dλ)2 |G| . This is a probability measure on b G, and for any central function, f = Z b G c f ∗(λ) Pl(λ) chλ . In particular, consider the normalized character of the regular representation of G: χCG (g) = δe,g. It has for Fourier transform the constant function equal to 1, since χCG = X λ∈b G Pl(λ)χλ . Thus, the Plancherel measure corresponds to the decomposition in normalized irreducible characters of the normalized regular trace of the group. The study of this probability measure in the case of symmetric groups will be the main topic of Chapter 13 of this book. Example. As an application of the results of this section, consider the following random process on the symmetric group S(n). We consider a deck of cards that are ordered from 1 to n, and at each time k ∈ N, we choose at random two independent indices i, j ∈ [ [1, n] ], and we exchange the i-th card of the deck with the j-th card of the deck, cards being counted from top to bottom. Each index i or j has probability 1 n , and it is understood that if i = j, then one leaves the deck of cards invariant. The configuration after k random transpositions of cards can be encoded by a permutation σk of size n, with σ(1) denoting the label of the first card of the deck, σ(2) denoting the label of the second card of the deck, etc. For instance, assuming n = 5, a possible trajectory of the process up to time k = 8 is k 0 1 2 3 4 5 6 7 8 σk 12345 15342 13542 43512 43512] 43521 23514 25314 25314 and there are two steps (k = 4 and k = 8) where the same index i = j was chosen. We denote P[A] the probability of an event A, and P[A| B] the probability of
- 47. 26 Representations of finite groups and semisimple algebras an event A conditionally to another event B. Consider the law fk(σ) = P[σk = σ], viewed as an element on CS(n). The rules of the random process are: P[σk+1 = τ|σk = σ] = ¨ 1 n if τ = σ (corresponding to choices of indices i = j); 2 n2 if τ = σ(i, j) for some pair i 6= j. Therefore, we get a recursion formula for fk: fk+1 = X τ∈S(n) P[σk+1 = τ]τ = X σ,τ∈S(n) fk(σ)P[σk+1 = τ|σk = σ]τ = X σ∈S(n) fk(σ)σ 1 n + 1 n2 X 1≤i6=j≤n (i, j) ! = fk 1 n + 1 n2 X 1≤i6=j≤n (i, j) ! . So, if f = 1 n + 1 n2 P 1≤i6=j≤n(i, j), then fk = f k for any k, in the sense of convolution in CS(n). It should be noticed that the recursion formula can be rewritten as fk+1 − fk = fk 1 n2 X 1≤i6=j≤n (i, j) − 1 ! . Thus, we are looking at the analogue in the setting of the symmetric group of the heat equation ∂ f ∂ t = 1 2 ∆f . Notice now that f is a linear combination of conjugacy classes in the symmet- ric group. Indeed, the identity 1 is a conjugacy class on its own, and on the other hand, two transpositions (i, j) and (k, l) are always conjugated: (k, l) = (i, k)(j, l)(i, j)(j, l)−1 (i, k)−1 and each transposition appears twice in P 1≤i6=j≤n(i, j). Moreover, since 1 = (1)−1 and (i, j) = (i, j)−1 , f ∗ = f . Therefore, f is in Z(CS(n)), and fk(σ) = X λ∈Ò S(n) Pl(λ) Ö (f k)∗(λ) χλ (σ) = X λ∈Ò S(n) Pl(λ) b f (λ) k χλ (σ). since the Fourier transform is an isomorphism of algebras. This can be rewritten as: fk(σ) = X λ∈Ò S(n) Pl(λ) 1 n + n − 1 n χλ (1,2) ‹k χλ (σ), since χλ (1) = 1 for any representation, and χλ (i, j) = χλ (1,2) for any transpo- sition (i, j) and any representation λ. This formula can be used to compute the asymptotics of the laws fk. In particular, it can be shown that −1 ≤ χλ (1,2) 1 if λ is not the trivial representation of S(n) on C. As a consequence, all the terms of
- 48. Semisimple algebras and modules 27 the previous formula go to zero as k grows to infinity, but the term corresponding to the trivial representation, which is (1)2 n! 1 n + n − 1 n ‹k 1 = 1 n! . So, limk→∞ fk(σ) = 1 n! , and the laws of the random process converge towards the uniform law on permutations. 1.4 Semisimple algebras and modules By Theorem 1.14, for any finite group G, the group algebra CG is isomorphic to a direct sum of matrix algebras L λ∈b G End(Vλ ), and if one endows CG and this sum of matrix algebras with adequate Hermitian structures, then one is able to do many computations on the group, e.g., to decompose any function in elemen- tary orthogonal components. Roughly speaking, the content of this section is the following: the same theory exists for any complex algebra that is isomorphic to a direct sum of matrix algebras, and moreover, there exists an abstract criterion in order to ensure that a given algebra is isomorphic to a direct sum of matrix algebras. There are many good reasons to consider this more general framework, and in this chapter, we shall see in particular that • it makes certain constructions on representations much more natural (in par- ticular, the induction of representations); • it allows one to develop a theory of duality between groups acting on a vector space (see Section 1.5). Later, it will also enable the study of combinatorial algebras that are modifications of the symmetric group algebras CS(n), and that are not group algebras. In this setting, we shall give and apply concrete criterions in order to ensure the semisim- plicity of the algebras considered. As a matter of fact, we will then also need to know the general representation theory of possibly non-semisimple algebras; this will be explained in Section 4.2, and the present section is an introduction to this more general theory. Â Algebras and modules. Though we mostly want to deal with algebras over C, it will be convenient in the beginning to consider algebras over an arbitrary field k. Thus, a field k being fixed, we recall that an algebra A over the field k is a k-vector space endowed with a product ×A : A× A → A that is
- 49. 28 Representations of finite groups and semisimple algebras (A1) associative and with a (unique) neutral element (the unity of the algebra): ∀a, b, c ∈ A, (a ×A b) ×A c = a ×A (b ×A c); ∃1A ∈ A, ∀a ∈ A, a ×A 1A = 1A ×A a = a. (A2) compatible with the external product of k: ∀λ ∈ k, ∀a, b ∈ A, λ(a ×A b) = (λa) ×A b = a ×A (λb). (A3) distributive with respect to the internal addition: ∀a, b, c ∈ A, a×A(b+c) = a×A b+a×Ac and (a+b)×Ac = a×Ac+b×Ac. In other words, a k-algebra is a ring and a k-vector space whose structures are compatible with one another in every possible way that one can think of. In the two first parts of this book, unless explicitly stated, we shall only work with finite- dimensional algebras, and denote dimk A, or simply dim A the dimension of A as a k-vector space. An algebra is said to be commutative if its product ×A is commu- tative: a ×A b = b ×A a. As before, we shall omit in most cases the product ×A and write a ×A b = ab. The properties listed above for an algebra ensure that this is a non-ambiguous notation. Example. Given a finite-dimensional k-vector space V, the set End(V) of k-linear maps u : V → V endowed with the product of composition of functions is a finite-dimensional algebra of dimension (dim V)2 . Similarly, the set of matrices M(n, k) of size n×n and with coefficients in k is an algebra for the matrix product. If n = dimk V, then the two algebras End(V) and M(n, k) are isomorphic, an isomorphism being given by u 7→ mat(e1,...,en)(u), where (e1,..., en) is an arbitrary basis of V. Example. For any finite group G, the set kG of formal k-linear combinations of elements of G (or, in other words, the set of functions G → k) is a k-algebra for the convolution product defined at the beginning of Section 1.3. It has dimension dimk kG = |G|. Example. For any field k, the set k[X1,..., Xn] of polynomials in n variables with coefficients in k is a commutative k-algebra. It is graded by the degree of polyno- mials, and this gradation is compatible with the algebra structure, meaning that for any elements a and b in the algebra, deg(ab) ≤ deg a + deg b. This inequality is an equality as soon as a and b are not zero.
- 50. Semisimple algebras and modules 29 A left module M over a k-algebra A is a k-vector space endowed with an external product · : A× M → M, such that (M1) · is compatible with the addition and the product in A: ∀a, b ∈ A, ∀m ∈ M, (a +A b) · m = (a · m) + (b · m); (a ×A b) · m = a · (b · m). (M2) · is compatible with the k-vector space structure on M: ∀a ∈ A, ∀m, n ∈ M, a · (m + n) = a · m + a · n; ∀a ∈ A, ∀λ ∈ k, ∀m ∈ M, λ(a · m) = (λa) · m = a · (λm). (M3) for all m ∈ M, 1A · m = m. Again, in the two first parts of this book, we shall only deal with finite-dimensional A- modules, and this assumption holds always implicitly in the following. Dually, one defines a right module M over a k-algebra A as a k-vector space endowed with an external product · : M ×A → M that is compatible with the structures of M and A. Notice that a k-vector space is a left module over the k-algebra k, so one can see the notion of module over a k-algebra as an extension of the notion of vector space over k. Then, the notions of (left or right) A-submodule and of morphism of (left or right) A-modules are defined in the obvious way, thereby generalizing the notions of k-vector subspace and of k-linear map. In the following, when a result holds for both left and right modules, we shall just speak of modules, and usually do the reasoning with left modules. Remark. Notice that a structure of A-module on a k-vector space M is equivalent to a morphism of k-algebras A → End(M). Example. Let V be a k-vector space. Then, V is a left module over End(V) for the operation u · v = u(v). Example. Let G be a finite group, and (V,ρ) a representation of G. Then, V is a left module over CG for the operation X g∈G f (g) g ! · v = X g∈G f (g)ρ(g)(v). Conversely, any left module V over CG is a representation of G for the rule ρ(g)(v) = g · v, the · denoting the product map CG × V → V. With this new point of view, a morphism between two representations V and W of G is a mor- phism of CG-modules. Therefore, there is an equivalence of categories between complex linear representations of G and left CG-modules. This reinterpretation already sheds a new light on certain results previously stated. For instance, the regular representation of G is an instance of the regular
- 51. 30 Representations of finite groups and semisimple algebras left module A associated to an algebra A, the action A× A → A being given by the product of the algebra. Then, in the second part of Lemma 1.5, the isomorphism of vector spaces between V and HomG(CG, V) comes from the more general fact that for any k-algebra A and any left A-module M, the module M is isomorphic as a k-vector space to HomA(A, M), the isomorphism being m 7→ (a 7→ a · m). If M is a left A-module, then M is also canonically a right Aopp -module, where Aopp denotes the k-algebra with the same underlying vector space as A, and with product a ×Aopp b = b ×A a. The right Aopp -module structure on a left A-module M is then defined by m·Aopp a = a·A m. In the case where A = CG, there is a simple realization of Aopp by using the inverse map. More precisely, CG → (CG)opp X g∈G f (g) g 7→ X g∈G f (g) g−1 is an isomorphism of C-algebras. Therefore, any left representation V of G admits a corresponding structure of right representation of G, given by v · g = g−1 · v. Example. The permutation representation of S(n) on Cn is more natural when given by a structure of right CS(n)-module. Indeed, it writes then as (x1, x2,..., xn) · σ = (xσ(1), xσ(2),..., xσ(n)). Similarly, consider a finite alphabet A, and denote the elements of An as words of length n with letters in A: An = {a1a2a3 ... an | ∀i ∈ [ [1, n] ], ai ∈ A}. There is a natural structure of right CS(n)-module on the space C[An ] of formal linear combinations of these words: (a1a2a3 ... an) · σ = aσ(1)aσ(2)aσ(3) ... aσ(n). There is also a natural structure of left CS(A)-module given by τ · (a1a2a3 ... an) = τ(a1)τ(a2)τ(a3)...τ(an). This kind of construction justifies the need of both notions of left and right mod- ules on an algebra. We shall study this double action more rigorously in Section 1.5, by introducing the notion of bimodule. Example. Let A be any k-algebra. Then A is both a left and right A-module, for the actions given by the product of the algebra. The left A-submodules of A are exactly the left ideals of A, and similarly on the right.
- 52. Semisimple algebras and modules 31 Â Semisimplicity and Artin–Wedderburn theorem. Let A be a k-algebra, and M be a (left) module over A that is not the zero mod- ule. The module M is said to be simple if it is of positive dimension and if its only submodules are {0} and M itself. It is said to be semisimple if it is a direct sum of simple modules. The notion of simple module is the generalization to the framework of modules and algebras of the notion of irreducible representation, and indeed, a representation V of G is irreducible if and only if it is a simple CG-module. Proposition 1.20. A finite-dimensional module M over a k-algebra A is semisimple if and only if, for every submodule N ⊂ M, there exists a complement A-submodule P with M = N ⊕ P. Proof. Suppose that M = Lr i=1 Mi is a direct sum of simple modules, and let N be a submodule of M. We take a subset I ⊂ [ [1, r] ] that is maximal among those such that N ∩ L i∈I Mi = {0}. By choice, if P = L i∈I Mi, then N + P is a direct sum. We claim that N ⊕ P = M. It suffices to show that for every i ∈ [ [1, r] ], N ⊕ P contains Mi. This is clear if i ∈ I. If i / ∈ I, then (N ⊕ P) ∩ Mi is not the zero submodule, since otherwise the set I would not be maximal. But Mi is simple, so (N ⊕ P)∩ Mi is a non-zero submodule of Mi, hence equal to the whole of Mi. This proves the existence of a complement A-module P of N such that M = N ⊕ P. Conversely, suppose that every submodule N of M has a complement sub- module P. We can exclude the trivial case M = {0}. Then, since M is finite- dimensional, there is no infinite descending chain of submodules of M, so M has necessarily a simple submodule M1. Denote P a complement of M1: M = M1 ⊕ P with M1 simple. To show that M is semisimple, it suffices now to prove that P has the same prop- erty as M, that is to say, that every submodule of P has a complement submodule in P. Indeed, an induction on the dimension of M will then allow us to conclude. Fix a submodule S ⊂ P. There is an isomorphism of A-modules ψ : P → M/M1 p 7→ [p]M1 . The A-submodule ψ(S) of M/M1 can be realized as the quotient module (M1 ⊕ S)/M1. By hypothesis, (M1 ⊕ S) has a complement R in M: M = (M1 ⊕ S) ⊕ R. Then, if πM1 : M → M/M1 is the canonical projection, πM1 (R) is a complement submodule of ψ(S) in M/M1, and T = ψ−1 (πM1 (R)) is a complement submodule
- 53. 32 Representations of finite groups and semisimple algebras of S in P since ψ is an isomorphism. M = (M1 ⊕ S) ⊕ R πM1 P = S ⊕ T oo ψ // M/M1 = ψ(S) ⊕ πM1 (R) Corollary 1.21. Semisimplicity of modules is kept by looking at submodules, quo- tient modules, and direct sum of modules. Proof. The stability by direct sum is trivial. For the two other properties, let M be a semisimple A-submodule, and P a submodule of M. We saw during the proof of Proposition 1.20 that P has the same property as M, so the stability for sub- modules is shown. On the other hand, if N is a complement of P in M, then the quotient M/P is isomorphic to N, which is a submodule of a semisimple module, hence semisimple; so the stability for quotient of modules is also proven. Definition 1.22. A finite-dimensional k-algebra A is said to be semisimple if every A-module M is semisimple. Proposition 1.23. A k-algebra A is semisimple if and only if the (left) A-module A is semisimple. Proof. If A is a semisimple algebra, then all its modules are semisimple, so A viewed as an A-module is semisimple. Conversely, suppose that Aviewed as a mod- ule is semisimple, and consider another finite-dimensional A-module M. Since M is finitely generated, it is isomorphic to a quotient of a module A ⊕ A ⊕ ··· ⊕ A. However, semisimplicity is kept for direct sums and quotients, so M is semisim- ple. We leave the reader to check that an easy consequence of this proposition and of Corollary 1.21 is that a quotient or a direct sum of semisimple k-algebras is again semisimple. We are now ready to classify the semisimple k-algebras. Recall that a division ring C over a field k is a (finite-dimensional) k-algebra such that for every non- zero c ∈ C, there exists b with bc = cb = 1. The difference with the notion of field extension of k is that we do not ask for the commutativity of the product in C. Given a division ring C, we denote M(n, C) the space of matrices with co- efficients in C; it is a (non-commutative) k-algebra for the product of matrices. A consequence of the possible non-commutativity of a division ring is that the multiplication on the left of C by C is not C-linear. Therefore, EndC (Cn ) is not k-isomorphic to the algebra M(n, C), but to the algebra M(n, Copp ), where Copp acts on C by multiplication on the right (this is C-linear). This subtlety appears in most of the following discussion.
- 54. Semisimple algebras and modules 33 Theorem 1.24 (Artin–Wedderburn). Every semisimple k-algebra A is isomorphic to M λ∈b A M(dλ, Cλ ) for some k-division rings Cλ and some multiplicities dλ ≥ 1. Lemma 1.25 (Schur). Let M be a simple A-module. Then, EndA(M) is a k-division ring. Proof. The kernel and the image of a morphism between A-modules are A- submodules. Thus, if M is a simple A-module and if u : M → M is a morphism of modules, then it is either 0 or an isomorphism. Hence, if u 6= 0, then u has an inverse v with uv = vu = idM . For the same reason, if M1 and M2 are two simple modules, then either they are isomorphic, or dimk HomA(M1, M2) = 0. Proof of Theorem 1.24. We decompose the left A-module A in a direct sum of sim- ple modules (ideals), gathered according to their classes of isomorphism as A- modules: A = M λ∈b A dλ Mλ , with dimHomA(Mλ , Mµ ) = 0 if λ 6= µ. For the moment, b A denotes the set of non-isomorphic simple modules appearing in A; we shall see hereafter that every simple module on A is isomorphic to some Mλ ∈ b A. We now use the following sequence of isomorphisms of k-algebras: Aopp = EndA(A) = M λ∈b A EndA(dλ Mλ ) = M λ∈b A M(dλ,EndA(Mλ )) = M λ∈b A M(dλ, Dλ ) where the Dλ are division rings. Let us detail each identity: 1. An endomorphism of left A-modules on A is necessarily ra : b 7→ ba for some a ∈ A. The composition of two endomorphisms reads then as ra1 ◦ ra2 = ra2a1 , so, a 7→ ra is an isomorphism of k-algebras between Aopp and EndA(A). 2. One has EndA(A) = HomA M λ∈b A dλMλ , M µ∈b A dµMµ ! = M λ,µ∈b A HomA(dλMλ , dµMµ ) = M λ∈b A EndA(dλ Mλ ) since two non-isomorphic simple modules Mλ and Mµ have no non-trivial morphism between them. These identities are a priori isomorphisms of k- vector spaces, but the two extremal terms are k-algebras, and it is easily seen that the identification between them is compatible with the product of com- position.
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- 56. exercised, continued unalterable through the several years in which I resided in Yarmouth, after my acquaintance with her commenced. I afterwards saw her several times during occasional visits which I made to that place, when I found that she still retained an affectionate remembrance of me.” She was in humble circumstances, and earned a scanty income by the use of her needle; but she coupled with it extraordinary efforts for the good of others, and this disposed some ladies, members of the Established Church, to contribute to her support. This enabled her to devote more time to her charitable work, and at length she was so absorbed in it that she became a kind of missionary to the inmates of the workhouse and the prisoners in Yarmouth gaol. She read and explained the Scriptures to them, and in devotional service, she carried on for their spiritual welfare, she employed parts of the Church Prayer-Book. Gradually, I infer, she became attached to those who helped her, and this association led to her becoming a member of the Establishment. After her death a commemorative window was placed in Yarmouth parish church, and at its reopening, after a costly restoration, Bishop Wilberforce pronounced an eloquent eulogium on Sarah Martin’s character. Some intimate Nonconformist friends of mine remained attached to her, and showed me numerous MSS. in her handwriting. I now return to the ranks of Dissent and proceed to notice— II. English Presbyterianism. A word on its earlier history will here be appropriate. The Presbyterians of the sixteenth and seventeenth centuries were orthodox. After the Restoration many of them adhered to the Westminster Confession, but a departure from it, in some instances, appeared in the century after. Arian and Socinian opinions began to obtain, but those who held them claimed connection with the Presbyterians of the Commonwealth, on the ground that they followed such worthies in the exercise of religious freedom and the rights of conscience. Their forefathers had repudiated the Prayer-Book, and now they, their sons in the cause of
- 57. religious freedom, renounced the Westminster Confession. For the most part they remained steadfast in believing New Testament miracles. The Rev. Mr. Madge, a noted English Presbyterian, sixty or seventy years ago, said to me once, he could not understand how a man could be called a Christian who did not believe in our Lord’s resurrection. During the reign of William IV. the two most prominent English Presbyterians of the old school were the Rev. Mr. Aspland and Mr. Madge. The latter I knew well. Mr. Aspland was an eloquent speaker, and exerted himself conspicuously in the cause of Unitarianism, with which he identified the interests of religious freedom. His son, in writing his father’s life, pourtrays that gentleman’s religious connections, social virtues, and decision of character; but does not conceal his warmth of temper, and dislike to certain eminent Trinitarians. Mr. Madge, before he became minister of Essex Street, London, was for some years settled in my native city, and presided over a wealthy congregation, in which were several distinguished literary and artistic people. The Martineaus, the Aldersons, the Starks, and other distinguished families, were of the number. They worshipped in the Octagon Chapel, as it was called from its architecture, and for a number of years the building was the most distinguished Nonconformist place of worship in the eastern capital. It was rather sumptuously fitted up in my boyish days, and the attendants were not wont to mix much with other Dissenters. If there were any fault in this, I dare say it was shared on both sides. Returning to the English Presbyterians at large, but especially as they existed in London, I must speak of a trust established by Dr. Williams, of the last century. He was orthodox, but the administration of funds bequeathed by him came into the hands of those Presbyterians who deviated from his doctrinal views, but still retained the Presbyterian name by which he was known. Though Unitarians in opinion, they by no means confined their charity to Unitarian ministers and chapels; and still the “Williams’ Scholarships”
- 58. are enjoyed by students preparing for orthodox ministrations amongst Independents. Dr. Martineau was for some time an administrator of the trust, but strongly objected to the exclusion of orthodox ministers from its administration. During the last century there were Presbyterians in England holding decidedly Evangelical views, and of late there have been numerous congregations gathered, which, in their unity, form what is called “The Presbyterian Church in England.” Scotch brethren of great renown—Dr. James Hamilton, Dr. Young, and Dr. Archer—I had the privilege of numbering amongst personal friends, and they were held in honour by all Evangelical Churchmen and Nonconformists. III. Another large section of brethren were Baptists, distinguished by certain doctrinal and disciplinary views;—the former as Particular or Calvinistic, on the one hand, and General or Arminian on the other;—the latter as Open communionists and Strict communionists. Open communionists admit to the Lord’s table those who have not been baptised by immersion; Strict communionists confine the Lord’s Supper to those who have been immersed. Such distinctions are now fading away. Calvinists and Arminians are comprehended in the same union, and Strict communionists are comparatively few. Robert Hall, the advocate of Open communion, I never saw: he died when I was young. Joseph Kinghorn, his opponent, a distinguished Hebrew scholar, I knew well, as he lived in Norwich during my boyhood. William Brock, who succeeded him, and afterwards became minister of Bloomsbury Chapel, London, entered the ministry about the same time as I did, and we regarded each other with warm affection. Dr. Cox and Dr. Steane were widely known in the religious world, and with both of them I entered into a fellowship of work and worship at the opening of chapels and on other public occasions. John Howard Hinton was another Baptist brother, of whom I saw much when he was at Reading and I was at Windsor. He was more original, more metaphysical, more scientific, and more excitable than others whom I have mentioned, perhaps of a higher intellectual order, and still greater depth of religious emotion. Mr.
- 59. Spurgeon, who has so recently left the world, and whose influence and fame extended further than any other Nonconformist in modern times, I greatly respected and admired; and though I did not share his intimacy, I saw something of him in my own home, and a little more in his, where he had a magnificent library, and received his numerous friends with cordiality. His popularity amongst aristocratic people was, for a little time, much greater than is generally supposed, for I was informed by a lady of distinction that for some weeks in his early career he was a leading topic of conversation in upper circles. IV. I now turn to the Quaker community. Well do I remember meetings at the Goldencroft, Norwich, where, at the upper end, sat men and women called Public Friends. My mother, born in 1770, told me of yearly meetings held in our old city, when sometimes Friends from America attended: and so great was the number of visitors that it raised the market price of provisions. Some ladies who came from the other side of the Atlantic wore dresses with open skirts and green aprons. No bows of ribbon were seen, while bonnets of black and of lead-coloured silk crowned the heads of young and old. What Charles Lamb says in his “Elia” corresponds with what I recollect, and what my mother used to tell me, how “troops of the shining ones” were seen walking the streets, on their way to the house of worship, where their silence was more eloquent than speech. I have read with sympathy “The Life of John Woolman,” written by himself, and so warmly recommended by the essayist. “Get,” says Charles Lamb, “the writings of John Woolman by heart, and love the early Quakers.” A very serious diversion in theological opinion existed among American Friends early in this century, and it is because an effect of it appeared in England that it is noticed here. A French Friend—the well-known Stephen Grellet—travelling in the States, makes this entry in his journal, under date 1822:—“We proceeded to Long Island, where I attended all the meetings, but here my soul’s distress exceeded all I had known during the preceding months,
- 60. though my baptism had been deep. I found that the greatest part of the members of our Society and many of the ministers and elders, are carried away by the principle which Elias Hicks has so assiduously propagated among them. He now speaks out boldly, disguising his sentiments no longer; he seeks to invalidate the Holy Scriptures, and sets up man’s reason as his only guide, openly denying the divinity of Christ. I have had many expostulations with him in which I have most tenderly pleaded with him, but all has been in vain.” [374] From what I have read in American literature touching what is known as the Hicksite controversy, it appears to me plainly indicative of a denial among many American Friends, that Jesus Christ, in the orthodox sense of the term, was Divine, and that He did not make any atonement for sin. Hicks appears to have been a thorough mystic, unintelligible to common-sense people. At all events he converted many to his views; and these views were caught up by some Friends in this country. To what extent exactly they were adopted in England I cannot say: but they created alarm amongst many Friends on this side the Atlantic. Great sorrow at the abandonment of Evangelical doctrines led to secessions from Quakerism on the part of excellent people who had been born and bred in the community. Some of them resided, at the time I speak of, on the borders of Wales, others in the county of York. They became Congregationalists, and in tours on behalf of the London Missionary Society, I was received hospitably in their homes, and had gratifying opportunities of witnessing their beautiful Christian life. Joseph John Gurney, of Earlham, felt seriously concerned respecting the American defection, in a community to which he had been attached from childhood. He had studied in the University of Oxford, had cultivated friendships in other denominations, was a good classic and Biblical scholar, and also an author of theological works. Mr. Gurney was “concerned” about the effect of Hicksite opinion on American and English Friends, and therefore took up his pen and wrote in reply to the leader who had done so much mischief.
- 61. Mr. Gurney, like his sister Mrs. Fry, undertook journeys for preaching the Gospel, and once he visited Windsor for that purpose. I was unwell at the time, but he called and talked by my bedside, and commended me to God in prayer. Several Quaker families at that period were living at Staines and Uxbridge; with them I had much intercourse, especially when we were joined in the advocacy of Slave Emancipation. The community, in both towns now named, was considerable for numbers and for wealth. Friends now dress, speak and act much like other people. Conforming to common custom, they still eschew all extravagances of fashion. They no longer forfeit membership by “marrying out of Society.” “The Right Honourable John Bright” (how shocked George Fox would have been at the title!) told me once, that relaxation in strictness as to unimportant points, had checked a decline in numbers going on before. V. Methodism, of course, brings to my mind a long train of early associations. Not merely names, but living forms, of noted preachers belonging to the second decade of this century come back to my recollection. Calvert Street Chapel was opened about 1812, and Dr. Coke preached. I cannot say that I remember his sermon; but, as noticed already, I distinctly recollect seeing the odd-looking, diminutive man, standing on a table talking in the committee room of Bethel Hospital [377] adorned by paintings of foundress and governors. Dr. Coke energetically addressed on the occasion a number of people, who had been invited by my grandfather, to hear the noted advocate of Methodist missions. Many years afterwards I mentioned the circumstance to a gentleman, who at the time took care of the patients, when he fetched an old committee book, in which this gathering was noticed, with a minute expressing the displeasure of the Governors at such a liberty being taken, and forbidding anything of the kind in future. The Wesleyan congregations in Norwich were
- 62. then very large, and local preachers—uncultivated men in humble life—frequently occupied the pulpit in the afternoon service at Calvert Street, and, remember, delivered animated discourses likely to do their hearers good. Dr. Jabez Bunting was a very influential man among the Methodists when I was young. For many years he was regarded as ruler of the Connexion,—exerting a despotic sway over the whole body. Such general conclusions oftentimes are not fairly drawn from existing facts, and how far widely extended opinion in the case now noticed, is justifiable I cannot undertake to say. To me he was very agreeable, and for him I had great respect. William Bunting, his son, was of a different stamp from his father, and though a skilful critic, he had not his father’s gift of authority and rule. Before the middle of the century came Dr. Newton, to open a second chapel, in the upper part of Norwich; his magnificent voice and careful diction produced a powerful effect. I met him in after-life at Windsor, when he told me that he was accustomed to leave his home on Monday morning in the Manchester circuit, and travel by coach to the other end of England,—perhaps cross over to Ireland,— and then get back, at the end of the week, ready for preaching the next day. He said he weekly delivered five or six sermons, making them “on the wheels” as he went along. He seemed a stranger to physical fatigue. During my Windsor ministry I became acquainted with a noted Wesleyan, who was not an itinerant, but a local, preacher. He went by the name of “Billy Dawson,” and was eminently gifted with humour and pathos. I heard him preach, and listened to his platform speeches. He was not only naturally eloquent, but histrionic too; in speeches and sermons he acted while he spoke. He made you realise what he described. It is said that George Whitefield, when preaching to sailors, described a storm at sea so vividly that some of them shouted, “Take to the long boat.” Dawson had a like power of realising what he described. He would, at a missionary meeting, make a telescope of his resolution, and putting
- 63. it to one of his eyes, describe what he saw in imagination,—perhaps a picture of the millennium drawn from Isaiah’s prophecies. I was young, just come from college, at the time I speak of, and made a speech in which I used some words which were not so plain as they might have been. After the meeting he spoke to me kindly, suggesting equivalent terms in plain Saxon. It was a good lesson for an unfledged bird. When I was a member of the Wesleyan Society, I attended class according to rule, and I found the practice beneficial, inasmuch as it was a constant spur to self-examination. The primitive agape, revived amongst the Methodists, exists under the name of love- feast, at which, together with eating bread and drinking water as an expression of fellowship, men and women are accustomed voluntarily to rise, and give some account of their religious experience for edification to others. These addresses I found often interesting and useful. By such means, a habit of spiritual intercommunication amongst Methodists is kept alive; beneficial in some cases no doubt, but liable to abuse in others, as most good things are. I am constrained to relate how this habit on the bright side manifested itself on a private occasion during a meeting of Conference in London. Dr. Jobson, an eminent Wesleyan, invited a party of friends to his house. He kindly included me in the number, and I found at his hospitable board the President for the year, and some ex-presidents. Together with them, Drs. Binney, Raleigh, Allon, and Donald Fraser were present. Our host was a thorough Methodist, and very comprehensive in his sympathies, for he had mixed with different denominations. He had many friends in the Establishment, and in early life had studied under an eminent Roman Catholic architect, at whose house he met bishops and priests of that communion. On the occasion I refer to, he in an easy way initiated a conversation which I can never forget. He appealed to his guests, one by one, for some account of their religious life. All readily responded; and this is most remarkable,—all who spoke attributed to Methodism spiritual influence of a decisive kind. To use Wesleyan phraseology, most of them had been “brought to God”
- 64. through Methodist instrumentality. Dr. Osborne was present, and made some remarks, at the close of which, with choked utterance, he repeated the verse— “And if our fellowship below, In Jesus be so sweet, What heights of rapture shall we know, When round the throne we meet?” The Norwich Methodists were chiefly humble folks with a sprinkling of some in better circumstances; their habits were very simple and they looked upon some who made money as becoming “worldly,” or at least, as exposed to temptation. At that time, however, such as possessed social comforts could not be justly charged with conformity to the course of this world; and over their little gatherings in one another’s houses there was shed a religious atmosphere such as was breathed in class and love-feast. Early in the century on a Sunday, between afternoon and evening service, there might be a large tea-party, where the preacher, a class-leader, and other members of Society would talk and pray and sing, till it was time to go to evening service at chapel. This communion seems to me now as I think of it such as is described in Malachi: “Then they that feared the Lord spake often one to another, and the Lord hearkened and heard it; and a book of remembrance was written before Him for them that feared the Lord and that thought upon His name; and they shall be Mine, saith the Lord of hosts, in that day when I make up My jewels, and I will spare them as a man spareth his own son that serveth him.” Worldly prosperity has since fallen to the lot of not a few Methodists, and the usual temptations surrounding wealth have tested their character; but I am thankful to say, amongst those whom I have visited, I have found beautiful instances of adherence to religious principles. I may mention a friend already noticed, Sir William McArthur, K.C.M.G. When Lord Mayor of London he continued his previous Wesleyan duties; and whilst bountiful in his hospitality
- 65. eschewed usages of a fashionable kind. In his year of office the Œcumenical Conference was held, and during its meetings repeated Mansion House invitations were given to friends in sympathy with Evangelical religion. I attended his funeral, and in his residence on Notting Hill a large number of mourners assembled, and we had a short devotional service together, very touching, tender, and beautiful. My personal recollections of Methodism, which roll back more than seventy years ago, linger round Yarmouth and Norwich. At Yarmouth I used to worship on a Sunday in a curious old-fashioned square chapel, with galleries on the four sides. There was a deep one opposite the two entrance doors, and attached to the front of that gallery was a pulpit—by what means, as a boy, I never could make out. The preacher ascended from behind by a staircase, invisible to the congregation, and then from the top of the staircase descended by two or three steps into a curiously shaped pulpit. I distinctly recollect the venerable Joseph Benson, then a patriarch, who had been associated with Methodists in John Wesley’s time. I think I see him now, of slender frame, venerable aspect, and wearing a coat of dark purple. Of course I have no recollection of what he said, but he was regarded as a saintly man in those days. In the autumn Yarmouth was frequented by a number of mariners from the north—coblemen they were called—who had come to fish for herrings off the Yarmouth coast. They were staunch Methodists, and used to hold a prayer-meeting after the general service. How those men used to pray with stentorian voice, which called forth loud “Amens” from voices all over the chapel! In Calvert Street, Norwich, there used to be special services on Christmas-day. After a prayer-meeting at six o’clock in the morning there was preaching at seven o’clock, when hymns appropriate to the season were sung, accompanied by violins and wind instruments of different kinds. I did not fail, between five and six o’clock, to rise and cross the city in order to be in good time for these services. They usually commenced with the hymn—
- 66. “Christians, awake, salute the happy morn Whereon the Saviour of mankind was born; Rise to adore the mystery of love, Which hosts of angels chanted from above; With them the joyful tidings first begun Of God incarnate and the Virgin’s son. “Then to the watchful shepherds it was told, Who heard the angelic herald’s voice: ‘Behold, I bring good tidings of a Saviour’s birth, To you and all the nations upon earth: This day hath God fulfilled His promised word, This day is born a Saviour, Christ the Lord.’” With the Methodist chapel in Calvert Street my earliest religious thoughts are connected. Watch-nights and love-feasts, are sacred in my recollection. VI. Respecting the Congregationalist denomination, of which I have spoken already, let me add that in 1877 I was requested by Dr. Schaff, of New York, to give my impression of prevalent beliefs amongst us. I replied as follows: “Looking at the principles of Congregationalism, which involve the repudiation of all human authority in matters of religion, it is impossible to believe that persons holding those principles can consistently regard any ecclesiastical creed or symbol in the same way as Catholics, whether Roman or Anglican, regard the creeds of the ancient Church. There is a strong feeling against the use of such documents for the purpose of defining limits of religious communion, or for the purpose of checking the exercise of free inquiry; and there is also a widespread conviction that it is impossible to reduce the expression of Christian belief to a series of logical propositions, so as to preserve and represent the full spirit of Gospel truth.” (See Schaff’s “Creeds of Christendom,” p. 833.) No doubt there may be heard in some circles loose conversation, seeming to indicate such a repugnance to creeds as would imply a
- 67. dislike to all formal definitions of Christian doctrine; but I apprehend the prevailing sentiment relative to this subject among our ministers and churches does not go beyond the point just indicated. Many of them consider that while creeds are objectionable as tests, and imperfect as confessions, they may have a certain value as manifestoes of conviction, on the part of different communities. Some people write and talk on the subject of present opinion, with a positiveness which only omniscience could warrant. No mortal can know what is going on in the minds of thousands, touching momentous subjects; yet such knowledge is requisite for the confident conclusions of certain critics. We may speak decidedly of what is commonly taught in a community, yet this should be done with qualifications and no farther. Silence on momentous points may prove a loss as to the full wealth of theology; but I am thankful for gain at the present day in richer views than formerly of our Lord’s character, and the bearing of it upon life and conduct. Let me add, however, if Redemption in all its fulness be not prominent in pulpit ministrations, power will be gone. Some suppose we are making theological advance, and that discoveries are opening akin to those in physical science; but people who have more carefully surveyed the wide field, and more observantly studied the history of religious thought, discover that much as seen at first sight, is chiefly a falling back upon what was old and forgotten. In closing what I have to say of modern Congregationalists, I venture to notice deceased ministers whom it has been a privilege to number amongst my friends. I knew but slightly the Rev. William Jay of Bath. He has been incidentally noticed in these pages already, for he was old when I was young. He rose from a lowly rank in life to be regarded as teacher and companion by the intellectual and noble. Mrs. Hannah More valued his ministrations and cultivated his society. Wilberforce used to attend his chapel when staying at Bath; and an Indian ruler,
- 68. when in England, went to hear him at Surrey Chapel, and expressed great admiration of the sermon. The next to be mentioned is John Angell James of Birmingham. I remember perfectly well the first sermon I heard him preach when I was a student. The text was: “Our conversation (or citizenship) is in heaven.” His voice was richly toned—a genuine birth gift improved by culture. He introduced the following illustration: A pilgrim in the Middle Ages, on his way to Jerusalem, passed through Constantinople. A friend took him from street to street, pausing to point out attractions, in magnificent buildings, and the rich scenery of the Golden Horn. He wondered the traveller was not enchanted. The latter replied: “Yes, all very fine, but it is not the Holy City.” The application was obvious and well enforced. Dr. Raffles of Liverpool—noticed already as one of my companions to Rome—and Dr. Hamilton of Leeds, well known throughout England, won the affections of their people by sympathetic intercourse, and interested them by eloquent instructions and appeals. The former enunciated his carefully prepared periods with a voice naturally musical, the latter delivered his thoughts in condensed sentences, which reminded one of a person taking very short steps. There was an intellectual power in the sermons of the last-named, not indicated in those of the former. John Alexander of Norwich I cannot pass by without notice. Like David, he was a youth with ruddy countenance. His speech throughout a sermon fell gentle as a snowflake, without any coldness of touch. He read much, and made good use of what he read. The charm of his private life and conversation exceeded the effect of his public ministry, though that was great. I must mention another name. John Harris was for some years a secluded pastor at Epsom, little known. He wrote “The Great Teacher,” but though far above the common level of such literature, it made little impression, compared with its merits. A prize was
- 69. offered for an essay on Covetousness and Christian Liberality. Harris won the prize, and printed the essay. The effect was instantaneous. The book sold edition after edition, and the author’s name became generally familiar. Requests for his services were universal. He was everywhere talked about, and when he preached places were crowded. His popularity lasted as long as he lived, but he died when he was fifty-four. He was unassuming, kind-hearted, generous to poor ministers, genial in conversation, and beloved by all who knew him. Another brother must be mentioned—Baldwin Brown—of superior intellectual type, well educated, an extensive reader, and one who delighted in a large circle of sympathetic friends. He gathered round him a good congregation, composed chiefly of thoughtful people, who became assimilated to his characteristic teachings. He wore himself out by incessant study and pulpit service. I must not pass by David Thomas of Bristol, my fellow-student and friend through life, whose elevated and genial character won from a wide circle warm attachment, and whose unique pulpit power captivated all capable of sympathising with one so thoughtful and so good. Nor can I omit Alexander Raleigh, my successor for a short period at Kensington, who fulfilled a ministry dear to many who listened with delight to his characteristic teaching. The last name I mention is that of Samuel Martin, minister at Westminster Chapel. He had gifts of a peculiar description, which marked him off, and made him stand by himself, both as minister and man. His appearance, voice, manner, habits, were all his own. He lived for his Church, in whose interests he was thoroughly absorbed. No one not intimately acquainted with him could have an adequate idea how he loved his flock, and lived for their welfare week by week. I had reverent affection for him as a saintly man, and I witnessed evidence amongst his large circle, in town and
- 70. country, how he watched for souls as one that must give an account. His congregation during Parliament months included several M.P.’s, whom he gathered together for patriotic prayer. His neighbour, Dr. Stanley, had a reverent regard for Mr. Martin, and I know that the Dean and Lady Augusta went to Westminster Chapel to hear his voice and worship with his people. He spoke to me of him in terms of strong affection, also telling me of a brother clergyman who, after a visit to his sick chamber, pronounced him one of the most saintly men he had ever seen. Printed by Hazell, Watson, Viney, Ld., London and Aylesbury.
- 71. FOOTNOTES [77] Faulkener’s “History and Antiquities of Kensington,” p. 317. [78] 1893. [80] “Christian Workers of the Nineteenth Century,” S.P.C.K., p. 216. [88a] “Life of E. B. Pusey,” i. 336. [88b] Ibid., ii. 33. [89] “Life of Pusey,” ii. 8. [126] Early Independent Churches had been particular in their relations to one another; and they would not recognise new communities without satisfactory evidence of character, principles, and conduct. They became more isolated afterwards. [176a] Now Archbishop of York. [176b] A very good account of this under the title of “Lectures on Bible Revision,” has been published by my excellent friend and late colleague at New College, Principal Newth, D.D. [183] “Memorials of a Quiet Life,” i. 237. [184] Dr. Raleigh, Sir Charles Reed, and others, were examined. [193] That was whilst I was in full work at Kensington, and not very long after our new chapel was built, while a debt of £1000 rested on it. I said I could not leave my charge whilst that debt remained. As soon as I had declined the New College principalship, my
- 72. congregation swept off the debt as expressive of gratitude for my remaining amongst them. [197] “Ecce Homo,” chap. iv. [230] Written about 1883. [233] I am glad that at Kensington, a liturgical element has been introduced, such as I should have approved, but could not accomplish, because I knew it would then be disapproved by many. [248] With a short Memoir by Robert Hall. [250] In what I have ventured to say about pulpit preparation I have hoped to help my younger ministerial brethren. [252] “Homes and Haunts of Martin Luther,” p. 4. [268] Since my visit to Ban de la Roche I discovered that, in a part of the country not far off, an Irish missionary, Columbanus, in the sixth century laboured for the temporal, as well as the spiritual, welfare of the people. See Wolf’s “Country of the Vosges,” p. 214. [315] Eusebius, “Eccl. Hist.,” V. i, 2. [316] Pastor and Madame Rodriguez. [318] De Aniccio, “L’Espagne traduit de Italien.” [329] “Life of Wilkie,” p. 472. [333] I have gone into this story in my “Spanish Reformers,” p. 185. [374] “Memoirs of Stephen Grellet,” vol. ii., 130. [377] See page 2.
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