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Landmarks in representation theory Gruson C. Digital
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Author(s): Gruson C., Serganova V
ISBN(s): 9782012042087, 2012042082
Edition: draft
File Details: PDF, 1.13 MB
Year: 2016
Language: english

Landmarks in representation theory
Caroline Gruson and Vera Serganova

Contents
Preface 7
Chapter 1. Introduction to representation theory of finite groups. 11
1. Definitions and examples 11
2. Ways to produce new representations 13
3. Invariant subspaces and irreducibility 14
4. Characters 17
5. Examples. 24
6. Invariant forms 28
7. Representations over R 31
8. Relationship between representations over R and over C 32
Chapter 2. Modules with applications to finite groups 35
1. Modules over associative rings 35
2. Finitely generated modules and Noetherian rings. 38
3. The center of the group algebra k (G) 40
4. One application 43
5. Generalities on induced modules 44
6. Induced representations for groups. 46
7. Double cosets and restriction to a subgroup 48
8. Mackey’s criterion 50
9. Hecke algebras, a first glimpse 50
10. Some examples 52
11. Some generalities about field extension 53
12. Artin’s theorem and representations over Q 54
Chapter 3. Representations of compact groups 57
1. Compact groups 57
2. Orthogonality relations and Peter–Weyl Theorem 65
3. Examples 68
Chapter 4. Some results about unitary representations 75
1. Unitary representations of Rn
and Fourier transform 75
2. Heisenberg groups and the Stone-von Neumann theorem 79
3. Representations of SL2 (R) 87
3

4 CONTENTS
Chapter 5. On algebraic methods 91
1. Introduction 91
2. Semisimple modules and density theorem 91
3. Wedderburn–Artin theorem 94
4. Jordan-Hölder theorem and indecomposable modules 95
5. A bit of homological algebra 99
6. Projective modules 103
7. Representations of artinian rings 108
8. Abelian categories 113
Chapter 6. Symmetric groups, Schur–Weyl duality and positive self-adjoint
Hopf algebras 115
1. Representations of symmetric groups 115
2. Schur–Weyl duality. 120
3. Generalities on Hopf algebras 124
4. The Hopf algebra associated to the representations of symmetric groups 127
5. Classification of PSH algebras part 1: decomposition theorem 129
6. Classification of PSH algebras part 2: unicity for the rank 1 case 131
7. Bases of PSH algebras of rank one 135
8. Harvest 141
9. General linear groups over a finite field 148
Chapter 7. Introduction to representation theory of quivers 159
1. Representations of quivers 159
2. Path algebra 162
3. Standard resolution and consequences 165
4. Bricks 169
5. Orbits in representation variety 171
6. Coxeter–Dynkin and affine graphs 173
7. Quivers of finite type and Gabriel’s theorem 177
Chapter 8. Representations of Dynkin and affine quivers 179
1. Reflection functors 179
2. Reflection functors and change of orientation. 181
3. Weyl group and reflection functors. 182
4. Coxeter functors. 183
5. Further properties of Coxeter functors 184
6. Affine root systems 186
7. Preprojective and preinjective representations. 189
8. Regular representations 191
9. Indecomposable representations of affine quivers 198
Chapter 9. Applications of quivers 201

CONTENTS 5
1. From abelian categories to algebras 201
2. From categories to quivers 204
3. Finitely represented, tame and wild algebras 208
4. Frobenius algebras 209
5. Application to group algebras 211
6. On certain categories of sl2-modules 214
Bibliography 227

Preface
Representation theory is a very active research topic in mathematics nowadays.
There are representations associated to several algebraic structures, representa-
tions of algebras, groups (of finite or infinite cardinal). Roughly speaking, a represen-
tation is a vector space equipped with a linear action of the algebraic structure. For
example, the algebra of n × n matrices acts on the vector space Cn
. A slightly more
complicated example is the action of the group GL(n, C) in the set of n×n-matrices,
the group acting by conjugation.
In the beginning, there was no tendency to classify all the representations of a
given object. The first result in this direction is due to Frobenius, who was interested
in the general theory of finite groups. Let G be a finite group, a representation V of
G is a complex vector space V together with a morphism of groups ρ : G → GL(V ).
One says V is irreducible if there exists no proper subspace W ⊂ V such that W
is stable under all ρ(g), g ∈ G. Frobenius showed there is finitely many irreducible
representations of G and that they are completely determined by their characters:
the character of V is the complex function g ∈ G 7→ Tr(ρ(g)) where Tr is the trace of
the endomorphism. These characters form a basis of the complex valued functions on
G invariant under conjugation. Then Frobenius proceeded to compute the characters
of symmetric groups in general. His results inspired Schur, who was able to relate
them to the theory of complex finite dimensional representations of GL(n, C) through
the Schur-Weyl duality. In both cases, every finite dimensional representation of the
group is a direct sum of irreducible representations (we say that the representations
are completely reducible).
The representation theory of symmetric groupes and the related combinatorics
turn out to be very useful in a lot of questions. We decided to follow Zelevinsky
and his book [37] and describe a Hopf algebra approach. This is an early example
of categorification, which was born before the fashionable term categorification was
invented.
Most of the results about representations of finite groups can be generalized to
compact groups. In particular, once more, the complex finite dimensional represen-
tations of a compact groups are completely reducible, and the regular representation
in the space of continuous functions on the compact group has the similar struc-
ture. This theory was developed by H. Weyl and the original motivation came from
quantum mechanics. The first examples of compact groups are the group SO(2)
7

8 PREFACE
of rotations of the plane (the circle) and the group SO(3) of rotations of the 3-
dimensional space. In the former case, the problem of computing the Fourier series
for a function on the circle is equivalent to the decomposition of the regular repre-
sentation. More generally, the study of complex representations of compact groups
helps to understand Fourier analysis on such groups.
If a topological group is not compact, for example, the group of real numbers
with operation of addition, the representation theory of such a group involves more
complicated analysis (Fourier transform instead of Fourier series). The representa-
tion theory of real non-compact groups was initiated by Harish-Chandra and by the
Russian school led by Gelfand. Here emphasis is on the classification of unitary rep-
resentations due to applications from physics. It is also worth mentioning that this
theory is closely related to harmonic analysis, and many special functions (such as
Legendre polynomials) naturally appear in the context of representation theory.
In the theory of finite groups one can drop the assumption that the characteristic
of the ground field is zero. This leads immediately to the loss of complete reducibility.
This representation theory was initiated by Brauer and it is more algebraic. If one
turns to algebras, a representation of an algebra is, by definition, the same as a
module over this algebra. Let k be a field. Let A be a k-algebra which is finite
dimensional as a vector space. It is a well-known fact that A-modules are not,
in general, completely reducible: for instance, if A = k[X]/X2
and M = A, the
module M contains kX as a submodule which has no A-stable complement. An
indecomposable A-module is a module which has no non-trivial decomposition as a
direct sum. It is also interesting to attempt a classification of A-modules. It is a
very difficult task in general. Nevertheless, the irreduducible A-modules are in finite
number. The radical R of A is defined as the ideal of A which annihilates each of
those irreducible modules, it is a nilpotent ideal. Assume k is algebraically closed,
the quotient ring A/R is a product of matrix algebras over k, A/R = ΠiEndk(Si)
where Si runs along the irreducible A-modules.
If G is a finite group, the algebra k(G) of k-valued functions on G, the composition
law being the convolution, is a finite dimensional k-algebra, with a zero radical as long
as the characteristic of the field k does not divide the cardinal of G. The irreducible
modules of k(G) are exactly the finite dimensional representations of the group G,
the action of G extends linearly to k(G). This shows that all k(G)-modules are
completely reducible (Maschke’s theorem).
In order to study finite dimensional k-algebras representations more generally,
it is useful to introduce quivers. Let A be a finite dimensional k-algebra, denote
S1, . . . , Sn its irreducible representations, and draw the following graph, called the
quiver associated to A: the vertices are labelled by the Sis and we put l arrows
between Si and Sj, pointing at Sj, if Ext1
(Si, Sj) is of dimension l (the explicit
definition of Ext1
requires some homological algebra which is difficult to summarize
in such a short introduction).

PREFACE 9
More generally, a quiver is an oriented graph with any number of vertices. Let Q
be a quiver, a representation of Q is a set of vector spaces indexed by the vertices of
Q together with linear maps associated to the arrows of Q. Those objects were first
systematically used by Gabriel in the early 70’s, and studied by a lot of people ever
since. The aim is to characterize the finitely represented algebras, or in other terms
the algebras with a finite number of indecomposable modules (up to isomorphism).
Today the representation theory has many flavors. In addition to the above
mentioned, one should add representations over non-archimedian local fields with its
applications to number theory, representations of infinite-dimensional Lie algebras
with applications to number theory and physics and representations of quantum
groups. However, in all these theories certain main ideas appear again and again
very often in disguise. Due to technical details it may be difficult for a neophyte to
recognize them. The goal of this book is to present some of these ideas in their most
elementary incarnation.
We will assume that the reader is familiar with usual linear algebra (including
the theory of Jordan forms and tensor products of vector spaces) and basic theory of
groups and rings.

CHAPTER 1
Introduction to representation theory of finite groups.
Beauty is the first test: there is no permanent place in the world for ugly mathe-
matics. (G.H. Hardy)
1. Definitions and examples
{defex}
Let k be a field, V be a vector space over k. By GL (V ) we denote the group of
all invertible linear operators in V . If dim V = n, then GL (V ) is isomorphic to the
group of invertible n × n matrices with entries in k.
A (linear) representation of a group G in V is a group homomorphism
ρ : G → GL (V ) ,
dim V is called the degree or the dimension of the representation ρ (it may be infinite).
For any g ∈ G we denote by ρg the image of g in GL (V ) and for any v ∈ V we
denote by ρgv the image of v under the action of ρg. The following properties are
direct consequences of the definition
• ρgρh = ρgh;
• ρ1 = Id;
• ρ−1
g = ρg−1 ;
• ρg (xv + yw) = xρgv + yρhw.
Example 1.1. (1) Let us consider the abelian group of integers Z with op-
eration of addition. Let V be the plane R2
and for every n ∈ Z, we set
ρn =

1 n
0 1

. The reader can check that this defines a representation of
degree 2 of Z.
(2) Let G be the symmetric group Sn, V = kn
. For every s ∈ Sn and (x1, . . . , xn) ∈
kn
set
ρs (x1, . . . , xn) = xs(1), . . . , xs(n)

.
In this way we obtain a representation of the symmetric group Sn which is
called the permutation representation.
(3) For any group G (finite or infinite) the trivial representation is the homo-
morphism ρ : G → k∗
such that ρs = 1 for all s ∈ G.
11

12 1. INTRODUCTION TO REPRESENTATION THEORY OF FINITE GROUPS.
(4) Let G be a group and
F (G) = {ϕ : G → k}
be the space of functions on G with values in k. For any g, h ∈ G and
ϕ ∈ F (G) let
ρgϕ (h) = ϕ (hg) .
Then ρ : G → GL (F (G)) is a linear representation.
(5) Recall that the group algebra k (G) is the vector space of all finite linear
combinations
P
cgg, cg ∈ k with natural multiplication. We define the
regular representation R : G → GL (k (G)) in the following way
Rs
X
cgg

=
X
cgsg.
Definition 1.2. Two representations of a group G, ρ : G → GL(V ) and σ :
G → GL(W) are called equivalent or isomorphic if there exists an invertible linear
operator T : V → W such that T ◦ ρg = σg ◦ T for any g ∈ G.
egularexample}
Example 1.3. If G is a finite group, then the representations in examples 4 and
5 are equivalent. Indeed, define T : F (G) → k (G) by the formula
T (ϕ) =
X
x∈G
ϕ (x) x−1
.
Then for any ϕ ∈ F (G) and g ∈ G we have
T (ρgϕ) =
X
x∈G
ρgϕ (x) x−1
=
X
x∈G
ϕ (xg) x−1
=
X
y∈G
ϕ (y) gy−1
= Rg (Tϕ) .
Let a group G act on a set X on the right. Let F(X) be the set of k-valued
functions on X. Then there is a representation of G in F(X) defined by
ρgϕ(x) := ϕ(x · g)
.
{ex1}
Exercise 1.4. Consider a left action l : G × X → X of G on X. For every
ϕ ∈ F(X), g ∈ G and x ∈ X set
σgϕ(x) = ϕ(g−1
· x).
(a) Prove that σ is a representation of G in F(X).
(b) Define a right action r : X × G → X by
x · g := g−1
· x,
and consider the representation ρ of G in F(X) associated with this action. Check
that ρ and σ are equivalent representations.
Remark 1.5. As one can see from the previous exercise, there is a canonical way
to go between right and left action and between corresponding representations.

2. WAYS TO PRODUCE NEW REPRESENTATIONS 13
2. Ways to produce new representations
Let G be a group.
Restriction. If H is a subgroup of G and ρ : G → GL (V ) is a representation
of G, the restriction of homomorphism ρ to H gives a representation of H which we
call the restriction of ρ to H. We denote by ResH ρ the restriction of ρ on H.
Lift. Let p : G → H be a homomorphism of groups. Then for every represen-
tation ρ : H → GL (V ), the composite homomorphism ρ ◦ p : G → GL (V ) gives a
representation of G on V . This construction is frequently used in the following case:
let N be a normal subgroup of G, H denote the quotient group G/N and p be the
natural projection. In this case p is obviously surjective. Note that in the general
case we do not require p to be surjective.
Direct sum. If we have two representations ρ : G → GL (V ) and σ : G →
GL (W), then we can define ρ ⊕ σ: G → GL (V ⊕ W) by the formula
(ρ ⊕ σ)g (v, w) = (ρgv, σgw) .
Tensor product. The tensor product of two representations ρ : G → GL (V )
and σ : G → GL (W) is defined by
(ρ ⊗ σ)g (v ⊗ w) = ρgv ⊗ σgw.
Exterior tensor product. Let G and H be two groups. Consider representa-
tions ρ : G → GL (V ) and σ : H → GL (W) of G and H respectively. One defines
their exterior tensor product ρ σ : G × H → GL (V ⊗ W) by the formula
(ρ σ)(g,h) v ⊗ w = ρgv ⊗ σhw.
{ex2}
Exercise 2.1. If δ : G → G × G is the diagonal embedding, show that for any
representations ρ and σ of G
ρ ⊗ σ = (ρ σ) ◦ δ.
Dual representation. Let V ∗
denote the dual space of V and h·, ·i denote the
natural pairing between V and V ∗
. For any representation ρ : G → GL (V ) one can
define the dual representation ρ∗
: G → GL (V ∗
) by the formula
hρ∗
gϕ, vi = hϕ, ρ−1
g vi
for every v ∈ V, ϕ ∈ V ∗
.
Let V be a finite-dimensional representation of G with a fixed basis. Let Ag for
g ∈ G be the matrix of ρg in this basis. Then the matrix of ρ∗
g in the dual basis of
V ∗
is equal to (At
g)−1
.
{ex3}
Exercise 2.2. Show that if G is finite, then its regular representation is self-dual
(isomorphic to its dual).

More generally, if ρ : G → GL (V ) and σ : G → GL (W) are two representations,
then one can naturally define a representation τ of G on Homk (V, W) by the formula
τgϕ = σg ◦ ϕ ◦ ρ−1
g , g ∈ G, ϕ ∈ Homk (V, W) .
{ex4}
Exercise 2.3. Show that if V and W are finite dimensional, then the represen-
tation τ of G on Homk (V, W) is isomorphic to ρ∗
⊗ τ.
Intertwining operators. A linear operator T : V → W is called an intertwining
operator if T ◦ ρg = σg ◦ T for any g ∈ G. The set of all intertwining operators will
be denoted by HomG (V, W). It is clearly a vector space. Moreover, if ρ = σ,
then EndG(V ) := HomG (V, V ) has a natural structure of associative k-algebra with
multiplication given by composition.
{ex5}
Exercise 2.4. Consider the regular representation of G in k(G). Prove that the
algebra of intertwiners EndG(k(G)) is isomorphic to k(G). (Hint: ϕ ∈ EndG(k(G))
is completely determined by ϕ(1).)
3. Invariant subspaces and irreducibility
3.1. Invariant subspaces and subrepresentations. Consider a representa-
tion ρ : G → GL (V ). A subspace W ⊂ V is called G-invariant if ρg (W) ⊂ W for
any g ∈ G.
If W is a G-invariant subspace, then there are two representations of G naturally
associated with it: the representation in W which is called a subrepresentation and the
representation in the quotient space V/W which is called a quotient representation.
{ex6}
Exercise 3.1. Let ρ : Sn → GL (kn
) be the permutation representation, then
W = {x(1, . . . , 1) | x ∈ k}
and
W0
= {(x1, . . . , xn) | x1 + x2 + · · · + xn = 0}
are invariant subspaces.
{ex7}
Exercise 3.2. Let G be a finite group of order |G|. Prove that any representation
of G contains an invariant subspace of dimension less or equal than |G|.
3.2. Maschke’s theorem.
{Maschke}
Theorem 3.3. (Maschke) Let G be a finite group such that char k does not divide
|G|. Let ρ : G → GL (V ) be a representation and W ⊂ V be a G-invariant subspace.
Then there exists a complementary G-invariant subspace, i.e. a G-invariant subspace
W0
⊂ V such that V = W ⊕ W0
.

3. INVARIANT SUBSPACES AND IRREDUCIBILITY 15
Proof. Let W00
be a subspace (not necessarily G-invariant) such that W ⊕W00
=
V . Consider the projector P : V → V onto W with kernel W00
. One has P2
= P.
Now we construct a new operator
P̄ :=
1
|G|
X
g∈G
ρg ◦ P ◦ ρ−1
g .
An easy calculation shows that ρg ◦ P̄ ◦ ρ−1
g = P̄ for all g ∈ G, and therefore ρg ◦ P̄ =
P̄ ◦ ρg. In other words, P̄ ∈ EndG(V ).
On the other hand, P̄|W = Id and Im P̄ = W. Hence P̄2
= P̄.
Let W0
= Ker P̄. First, we claim that W0
is G-invariant. Indeed, let w ∈ W0
,
then P̄ (ρgw) = ρg P̄w

= 0 for all g ∈ G, hence ρgw ∈ Ker P̄ = W0
.
Now we prove that V = W ⊕ W0
. Indeed, W ∩ W0
= 0, since P̄|W = Id. On the
other hand, for any v ∈ V , we have w = P̄v ∈ W and w0
= v − P̄v ∈ W0
. Thus,
v = w + w0
, and therefore V = W + W0
.
Remarks. If char k divides |G| or G is infinite, the conclusion of Mashke’s
theorem does not hold anymore. Indeed, in the example of Exercise 3.1 W and W0
are complementary if and only if char k does not divide n. Otherwise, W ⊂ W0
⊂ V ,
and one can show that neither W nor W0
have a G-invariant complement.
In the case of an infinite group, consider the representation of Z in R2
as in the
first example of Section 1. The span of (1, 0) is the only G-invariant line. Therefore
it can not have a G-invariant complement in R2
.
3.3. Irreducible representations and Schur’s lemma.
Definition 3.4. A non-zero representation is called irreducible if it does not
contain any proper non-zero G-invariant subspace.
{exdimir}
Exercise 3.5. Show that the dimension of any irreducible representation of a
finite group G is not bigger than its order |G|.
The following elementary statement plays a key role in representation theory.
Lemma 3.6. (Schur) Let ρ : G → GL(V ) and σ : G → GL(W) be two irreducible {Schur}
representations. If T ∈ HomG(V, W), then either T = 0 or T is an isomorphism.
Proof. Note that Ker T and Im T are G-invariant subspaces of V and W, re-
spectively. Then by irreducibility of ρ, either Ker T = V or Ker T = 0, and by
irreducibility of σ, either Im T = W or Im T = 0. Hence the statement.
{corschur}
Corollary 3.7. (a) Let ρ : G → GL(V ) be an irreducible representation. Then
EndG(V ) is a division ring.
(b) If the characteristic of k does not divide |G|, EndG(V ) is a division ring if and
only if ρ is irreducible.
(c) If k is algebraically closed and ρ is irreducible, then EndG(V ) = k.

Proof. (a) is an immediate consequence of Schur’s Lemma.
To prove (b) we use Maschke’s theorem. Indeed, if V is reducible, then V = V1⊕V2
for some proper subspaces V1 and V2. Let p1 be the projector on V1 with kernel V2 and
p2 be the projector onto V2 with kernel V1. Then p1, p2 ∈ EndG(V ) and p1 ◦ p2 = 0.
Hence EndG(V ) has zero divisors.
Let us prove (c). Consider T ∈ EndG(V ). Then T has an eigenvalue λ ∈ k and
T −λ Id ∈ EndG(V ). Since T −λ Id is not invertible, it must be zero by (a). Therefore
T = λ Id.
3.4. Complete reducibility.
Definition 3.8. A representation is called completely reducible if it splits into a
direct sum of irreducible subrepresentations. (This direct sum might be infinite.)
{cr}
Theorem 3.9. Let ρ : G → GL(V ) be a representation of a group G. The
following conditions are equivalent.
(a) ρ is completely reducible;
(b) For any G-invariant subspace W ⊂ V there exists a complementary G-
invariant subspace W0
.
Proof. This theorem is easier in the case of finite-dimensional V . To prove it for
arbitrary V and G we need Zorn’s lemma. First, note that if V is finite dimensional,
then it always contains an irreducible subrepresentation. Indeed, we can take a
subrepresentation of minimal positive dimension. If V is infinite dimensional then
this is not true in general.
subinvariance}
Lemma 3.10. If ρ satisfies (b), any subrepresentation and any quotient of ρ also
satisfy (b).
Proof. To prove that any subrepresentation satisfies (b) consider a flag of G-
invariant subspaces U ⊂ W ⊂ V . Let U0
⊂ V and W0
⊂ V be G-invariant subspaces
such that U ⊕ U0
= V and W ⊕ W0
= V . Then W = U ⊕ (U0
∩ W).
The statement about quotients is dual and we leave it to the reader as an exercise.

{minirr}
Lemma 3.11. Let ρ satisfy (b). Then it contains an irreducible subrepresentation.
Proof. Pick up a non-zero vector v ∈ V and let V 0
be the span of ρgv for all
g ∈ G. Consider the set of G-invariant subspaces of V 0
which do not contain v, with
partial order given by inclusion. For any linearly ordered subset {Xi}i∈I there exists
a maximal element, given by the union
[
i∈I
Xi. Hence there exists a proper maximal
G-invariant subspace W ⊂ V 0
, which does not contain v. By the previous lemma one
can find a G-invariant subspace U ⊂ V 0
such that V 0
= W ⊕U. Then U is isomorphic
to the quotient representation V 0
/W, which is irreducible by the maximality of W in
V 0
.

4. CHARACTERS 17
Now we will prove that (a) implies (b). We write
V =
M
i∈I
Vi
for a family of irreducible G-invariant subspaces Vi. Let W ⊂ V be some G-invariant
subspace. By Zorn’s lemma there exists a maximal subset J ⊂ I such that
W ∩
M
j∈J
Vj = 0.
We claim that W0
:=
M
j∈J
Vj is complementary to W. Indeed, it suffices to prove that
V = W + W0
. For any i /
∈ J we have (Vi ⊕ W0
) ∩ W 6= 0. Therefore there exists
a non-zero vector v ∈ Vi equal to w + w0
for some w ∈ W and w0
∈ W0
. Hence
Vi ∩ (W0
+ W) 6= 0 and by irreducibility of Vi, we have Vi ⊂ W + W0
. Therefore
V = W + W0
.
To prove that (b) implies (a) consider the family of all irreducible subrepresen-
tations {Wk}k∈K of V . Note that
X
k∈K
Wk = V because otherwise
X
k∈K
Wk has a
G-invariant complement which contains an irreducible subrepresentation. Again due
to Zorn’s lemma one can find a minimal J ⊂ K such that
X
j∈J
Wj = V . Then clearly
V =
M
j∈J
Wj.
The next statement follows from Maschke’s theorem and Theorem 3.9.
{cormaschke}
Proposition 3.12. Let G be a finite group and k be a field such that char k does
not divide |G|. Then every representation of G is completely reducible.
4. Characters
{characters}
4.1. Definition and main properties. For a linear operator T in a finite-
dimensional vector space V we denote by tr T the trace of T.
For any finite-dimensional representation ρ : G → GL (V ) the function χρ : G → k
defined by
χρ (g) = tr ρg.
is called the character of the representation ρ.
{charex}
Exercise 4.1. Check the following properties of characters.
(1) χρ (1) = dim ρ;
(2) if ρ ∼
= σ, then χρ = χσ;
(3) χρ⊕σ = χρ + χσ;
(4) χρ⊗σ = χρχσ;
(5) χρ∗ (g) = χρ (g−1
);

(6) χρ (ghg−1
) = χρ (h).
{charperm}
Exercise 4.2. Calculate the character of the permutation representation of Sn
(see the first example of Section 1).
{charreg}
Example 4.3. If R is the regular representation of a finite group, then χR (g) = 0
for any g 6= 1 and χR (1) = |G|.
Example 4.4. Let ρ : G → GL (V ) be a representation of dimension n and
assume char k 6= 2. Consider the representation ρ⊗ρ in V ⊗V and the decomposition
V ⊗ V = S2
V ⊕ Λ2
V.
The subspaces S2
V and Λ2
V are G-invariant. Denote by sym and alt the subrepre-
sentations of G in S2
V and Λ2
V respectively. Let us compute the characters χsym
and χalt.
Let g ∈ G and denote by λ1, . . . , λn the eigenvalues of ρg (taken with multiplic-
ities). Then the eigenvalues of altg are the products λiλj for all i j while the
eigenvalues of symg are λiλj for i ≤ j. This leads to
χsym (g) =
X
i≤j
λiλj,
χalt (g) =
X
ij
λiλj.
Hence
χsym (g) − χalt (g) =
X
i
λ2
i = tr ρg2 = χρ g2

.
On the other hand by properties (3) and (4)
χsym (g) + χalt (g) = χρ⊗ρ (g) = χ2
ρ (g) .
Thus, we get
(1.1) χsym (g) =
χ2
ρ (g) + χρ (g2
)
2
, χalt (g) =
χ2
ρ (g) − χρ (g2
)
2
.
{complexcase}
Lemma 4.5. If k = C and G is finite, then for any finite-dimensional representa-
tion ρ and any g ∈ G we have
χρ(g) = χρ(g−1).
Proof. Indeed, χρ(g) is the sum of all the eigenvalues of ρg. Since g has finite
order, every eigenvalue of ρg is a root of 1. Therefore the eigenvalues of ρg−1 are the
complex conjugates of the eigenvalues of ρg.

4. CHARACTERS 19
{orthrel}
4.2. Orthogonality relations. In this subsection we assume that G is finite
and the characteristic of the ground field k is zero. Introduce a non-degenerate
symmetric bilinear form on the space of functions F (G) by the formula
(1.2) (ϕ, ψ) =
1
|G|
X
s∈G
ϕ s−1

ψ (s) .
If ρ : G → GL(V ) is a representation, then we denote by V G
the subspace of
G-invariant vectors, i.e.
V G
= {v ∈ V |ρg(v) = v, ∀g ∈ G}.
{orth1}
Lemma 4.6. If ρ : G → GL(V ) is a representation, then
dim V G
= (χρ, χtriv),
where χtriv denotes the character of the trivial representation, i.e. χtriv(g) = 1 for all
g ∈ G.
Proof. Consider the linear operator P ∈ EndG(V ) defined by the formula
P =
1
|G|
X
g∈G
ρg.
Note that P2
= P and Im P = V G
. Thus, P is a projector on V G
. Since char k = 0
we have
tr P = dim Im P = dim V G
.
On the other hand, by direct calculation we get tr P = (χρ, χtriv), and the lemma
follows.
Note that for two representations ρ : G → GL(V ) and σ : G → GL(W) we have
(1.3) Homk(V, W)G
= HomG(V, W) = (V ∗
⊗ W)G
.
Therefore we have the following
{orth2}
Corollary 4.7. One has
dim HomG(V, W) = (χρ, χσ).
Proof. The statement is a consequence of the following computation:
(χρ, χσ) =
1
|G|
X
g∈G
χρ(g−1
)χσ(g) =
1
|G|
X
g∈G
χρ∗⊗σ(g) = (χρ∗⊗σ, χtriv).

The following theorem is usually called the orthogonality relations for characters.

{orth}
Theorem 4.8. Let ρ, σ be irreducible representations over a field of characteristic
zero.
(a) If ρ : G → GL(V ) and σ : G → GL(W) are not isomorphic, then (χρ, χσ) = 0.
(b) Assume that the ground field is algebraically closed. If ρ and σ are equivalent,
then (χρ, χσ) = 1.
Proof. By Schur’s lemma
HomG(V, W) = 0.
Therefore Corollary 4.7 implies (a).
Assertion (b) follows from Corollary 3.7 (c) and Corollary 4.7.
This theorem has several important corollaries.
{orth3}
Corollary 4.9. Let
ρ = m1ρ1 ⊕ · · · ⊕ mrρr
be a decomposition into a sum of irreducible representations, where miρi is the direct
sum of mi copies of ρi. Then mi =
(χρ,χρi )
(χρi ,χρi )
.
The number mi is called the multiplicity of an irreducible representation ρi in ρ.
{orth7}
Corollary 4.10. Two finite-dimensional representations ρ and σ are equivalent
if and only if their characters coincide.
In the rest of this section we assume that the ground field is alge-
braically closed.
{orth4}
Corollary 4.11. A representation ρ is irreducible if and only if (χρ, χρ) = 1.
{exttens}
Exercise 4.12. Let ρ and σ be irreducible representations of finite groups G and
H respectively.
(a) If the ground field is algebraically closed, then the exterior product ρ σ is
an irreducible representation of G × H.
(b) Give a counterexample to (a) in the case when the ground field is not alge-
braically closed.
{orth5}
Theorem 4.13. Every irreducible representation ρ appears in the regular repre-
sentation with multiplicity dim ρ.
Proof. The statement is a direct consequence of the following computation
(χρ, χR) =
1
|G|
χρ (1) χR (1) = dim ρ.

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you are located before using this eBook.
Title: The other half
Author: Edwin L. Sabin
Release date: May 6, 2024 [eBook #73550]
Language: English
Original publication: Indianapolis, IN: Popular Fiction Publishing
Company, 1926
Credits: Roger Frank and Sue Clark
*** START OF THE PROJECT GUTENBERG EBOOK THE OTHER
HALF ***

“Skull and skeleton lay eery and mysterious, whitely
gleaming, bleached by many weathers.”
The Other Half
By Edwin L. Sabin
I had advertised that a passenger would be taken (for a price) on my
return trip by air from Omaha to the coast, and awaited the
responses with no little curiosity. A flying companion should be
chosen carefully.
The very first applicant, therefore, startled me. He appeared
almost as soon as the papers were off the presses—a spare,
intense, elderly man, with gray mustache and imperial, and bushy
brows shadowing singularly bright, restless eyes. His years, of
course, were against him; his weathered, lean face and active step
bespoke energy, nevertheless I judged him to be rising sixty.
“But, my dear sir!” I protested.
He proved insistent.

“Heart perfect,” he snapped. “I’ll leave that to any doctor you
appoint. Heart, arteries and lungs, they’re as sound as yours.”
“Just why do you wish to make the trip, may I ask?” said I.
“Business or pleasure?”
“Business.” He eyed me sharply. “There’ll be no woman aboard?”
“Scarcely,” I assured.
“All right. No woman. I’ve been across time and again, by train—
and there were women; by auto—drove my own car, alone, but there
were the women, before, behind, and no way to avoid them.” He
grumbled almost, savagely. “I’ll go by air,” he resumed. “I want to get
to San Francisco at once. I want to look around. Do you stop at
Denver? Salt Lake? Cheyenne?”
“Straight to San Francisco,” said I. “We may have to land en route,
perhaps Cheyenne, perhaps Reno; but not for long and I make no
promises.”
“All right. I’ll look around San Francisco. I may have missed
something. Then I can work back. I’m not through. You’ll have to take
me. I’ll pay you double. I’m sounder than most of the younger men; I
have no family——”
“You’re not married, sir?” I queried.
“No, no! Thank God, no! You accept me?”
He noted me hesitate. Perhaps he sensed that I deemed him a
trifle off center.
“I’ll give you references,” he proffered with dignity. “I’m not crazy—
not quite. Look me up, for I mean to go. San Francisco, again: then I
can work back. There’s always the chance,” he muttered. “Yes,
there’s always the chance.” And he challenged: “If you find me sane
and sound, it’s a bargain, is it?”
“Possibly so, in case——”
“And we start at once?”
“Tomorrow.”
He paused.
“You’re a Westerner?”
“Born and raised in Leadville, Colorado,” I assured.
He seized upon the fact.
“Ah! Leadville! We couldn’t stop there?”

“Hardly.”
“But it was a busy camp, once, wasn’t it? A typical camp; a
rendezvous, with dance halls, gambling dens, and men and women
of all kinds gathered?”
“A boom camp, and wide open,” I said. “That was before my day,
however.”
“Yes,” he pursued. “So it was. I’ve been there. I must look into it
again. It’s one more place. You were born and raised there, you say?
Lived there some time? Wait! Did you ever happen to see the mate
to this, in curiosity shops, say, or among relics of the old-timers?”
Thereupon he unsnapped a small protective leather case and
passed me the half of a silver coin, pierced as if it once had been
strung.
“An old half dollar?” I hazarded.
“Yes. If you’ve aviator’s eyes you can read the lettering around the
rim, young man.”
So I could. “God Be With You——” was the legend, unfinished as if
cut short. He was gazing anxiously at me, his lips a-tremble. I turned
the piece over and passed it back.
“No,” said I; “I never happened to see the other half. A keepsake?”
His face set sternly. He restored the half coin to its case.
“A keepsake. You are married, young man?”
“Not yet.”
“Don’t,” he barked. “Don’t. Pray God you may be spared that.”
A woman-hater, he; odd in a man who should be mellowing. But
upon looking him up I found that this was his only apparent
defection. A strange, restless man, however, with few friends;
antecedents unknown; personal history taboo with him; and
wanderlust possessing him today as yesterday and the day before.
“Again?” his banker blurted. “Bound across again? He only just got
back from San Francisco, by automobile, via Salt Lake, Cheyenne
and Denver. Drove alone. So he’s going through with you? That’ll be
his fifth or sixth trip this year. He’s a regular Wandering Jew.”
“And his business?” I invited.
“Business? None.”
“On the trips, I mean.”

“My dear man, nobody knows. He goes and comes, goes and
comes. You’d think he was hunting a lost mine; or a lost child, only
he says he isn’t married. I believe he has covered the West from end
to end and border to border. Did he show you his pocketpiece?”
“A half coin? Yes. And asked me if I’d ever seen the other half.”
“That’s it. He asks everybody the same, especially if they’re
Western people. What he wants of the other half, no one knows. A
fad, maybe; an excuse to keep moving. He’ll not find it in the air,
that’s certain.”
“Not in the air,” I agreed. “He must have other reasons for going by
that route. To avoid women, he intimated.”
“And to get there quickly. He never comes home satisfied. No
sooner gets here than something seems to call him; you’d think he
had an S. O. S. wireless by the way he hustles out again, maybe
over the very same trail. Always searching, always searching; that’s
the life of old John. And never finding.”
“He’s past sixty?” I asked.
“Past sixty! He’s past seventy, but nobody’d guess it.”
And I accepted John Brown as passenger. No one else offered as
likely. I notified him to be ready. We hopped off in the morning.
The iron rails crushed the romance of plains travel. With the airplane
also the crossing of the West, like the crossing of the East, is
business. In the long overland stretches the aviator pays scant
attention to the dead epics that he violates with the drumming blast
of his propeller when he bores through the atmosphere above those
plains where the spirits still dance in little dust whirls that pivot and
career with no breath of wind. But I’ve often wandered what
imploring shades we dislocate when we ride in that half-world ether
which is neither heaven nor earth.
My passenger and I made our first leg without event. Out of
Cheyenne the motor began to buck, and a rudder control jammed
annoyingly. There was only one thing to do. Spiraling and slanting
like a wing-tipped bird we sought a landing place.
The country below, as revealed, was rugged, inhospitable desert—
a bad-lands desert with deeply graven face upturned immutable.

Plunging from high covert as we did, and bursting into full earth-view,
we should have appeared like a prodigy from the nethermost. But no
buffalo rocked in flight, no antelope scoured flashily, no red warriors
hammered their ponies for refuge. I saw, however, far, far away
toward the horizon, the smoke thread of a train, and I read in the
signal a message of derision.
We skimmed above a flattish uplift. Fissures and canyons yawned
for us. My passenger’s voice dinned hollowly into my ear, through
our ’phone.
“A country God forgot. And there’s nothing here. Useless, useless!
We must go on.”
But I had to do it. Passing with a great rush we turned into the
wind, and breasting, fluttering, managed to strike just at the edge of
a flat-top butte or mesa. We bounded, rolled, checked, halted, and
there we were.
My passenger was out first, divested of his safety harness. He
acted like one distraught. Our brief stop near Cheyenne had vexed
him—he had wished to spend either more time there, or less time.
Now this impromptu stop enraged him.
“What a place, what a place!” he stormed. “There’s nothing here;
there can be nothing here. We must get on. I’m wasting time. I paid
to get on, to San Francisco; even Salt Lake. Then I can work back.
But what am I to do here? And I’m growing old. How long will you
be?”
“Not long. And meanwhile,” I retorted, “you’ll not be bothered with
women. You can be thankful for that.”
He snorted.
“Women! No women here; yes. A spot without woman: man and
God. We’ve got to get on. I’ll pay you well to get me on. Do you
hear? To San Francisco—to Salt Lake; some center where I can
look, look, and then work back. I must look again.”
He strode frenziedly. A glance about as I stripped myself of
incumbrances showed me that we were isolated. The mesa dropped
abruptly on all sides; by a running start we might soar from an edge
like a seaplane from the platform of a battleship. And I noted also
that without doubt we should have to depend upon our own

resources, for if this was a country God forgot it moreover seemed to
be a country by man forgotten, granted that man ever before had
known it. All furrowed and washed and castlemented, it was a region
where we might remain pancaked and unremarked, as insignificant
as a beetle.
I was hunting our engine trouble, when on a sudden he called, and
beckoned.
“Here, you! What’s this?”
I went over. Something quickened me, electric and prickling as
when one’s flesh crawls in contact with a presence unseen. Skull
and skeleton lay eery and mysterious, whitely gleaming, bleached by
many weathers. He stooped——
“Great God!” he stammered.
“You’ve found it?” I asked: and I knew that he had, even while he
was polishing it against his sleeve.
“I don’t know, I don’t know. Look at it. Tell me. I can’t see. What is
it?”
His hand shook as with palsy as he extended it to me; then the
half of a silver coin, plucked from the loosened grip of skeleton
fingers; the date——
“Give it to me,” he cried, and snatched at it.
The date, 1866; and the legend, upon the side less tarnished, “—
Till We Meet Again.” He fumbled in his pocket. The two halves
matched sufficiently—“God Be With You Till We Meet Again.”
“What you’ve been looking for?” I prompted.
He stared dazedly at me.
“Looking for! A thousand times. A thousand years. No, no; not that
long, but more than fifty years. Denver, Cheyenne, Salt Lake, San
Francisco, Los Angeles, Helena, Laramie, Creede, Deadwood,
Leadville, Dodge City—wherever men and women of her kind
gathered in her day and his I’ve searched again and again. Not for
her! She must be dead, and long dead. But for word of this; for this,
or trace of this. It was mine. I gave it. And now, here! How came it
here? Those bones won’t speak.” He angrily kicked them. “Speak!

What were you doing with this half coin? Where was she? Were you
man or woman?”
“Woman, Mr. Brown,” said I.
His jaw tautened as he faced me full.
“You say woman? How do you know? What woman?”
“I know,” said I. “And what woman? A young woman, a girl,
somebody’s wife who was supposed to have run away with a breed
on the Overland Trail fifty odd years ago—but didn’t.”
He recoiled a step, tottering, countenance blanched.
“What? Supposed! Supposing I say there was such a woman—my
own wife, sir—my bar sinister—my cross that has ruined my life and
made me doubt God and man and woman for half a century. And
this half coin! I vowed I’d have it back. When at old Fort Bridger I got
word that she had deserted me—deserted me for a scoundrelly half-
breed—I swore that I’d trail her down till I got back the only bond
between us. It’s been my passion; it’s been something to live for.
That was 1867; this is 1920. I am seventy-four years of age. I have
covered the West, and cursed women while cursing her. And to what
end? This forsaken spot, a mess of bones, and no word! Oh, God! I
thought I didn’t care—she deserved the worst that could happen to
her. This is the keepsake token. Yes. But where is she? I loved her. I
want to know.”
He shut his face in his quivering hands.
I put my hand upon his shoulder.
“Come, come,” said I. “The half of the coin and the half of the story
have been yours. Shall I tell you the other half of the story, to match
this other half of the coin? It says ‘Till We Meet Again’, remember.”
Then he faced me once more.
“We halved the coin when we parted in the States, I for Fort
Bridger as a government clerk there, she to wait till I should send for
her. Yes, yes. Fifty and more years ago. ‘Till We Meet Again’! And
mine: ‘God Be With You’! Ah! What do you know? How can you
stand and tell me of her? Did you ever see her—did you ever see
her?” He clutched me by the arm. “Did you ever see her, that hussy,
that scarlet woman, that—that—yes, and my own wife who made me
lose faith in woman, man, and God; took my youth from me, sent me

wandering about without home and without charity? Curse her! The
end of the trail, and what do I find? Dry bones. Whose bones?” He
faltered, and he implored, simply: “You guessed? You’re too young to
have been on the plains in those days. Did you know him?”
“Pierre Lavelle?”
“Ah!” he quavered. He dashed down the half coin. “Are you going
to tell me these bones are his? No, no! Such men as he live long.
And this keepsake! Tell me she died miserably; that will be
something. You did know him? You did? Or do you dare to allege
you can rebuild a past, from this dungheap? What?”
“You wrong her, Mr. Brown,” I answered. “I never knew Lavelle,
never saw him, I never knew her—I do not even know her name,
except by yours. But——”
“Catherine,” he murmured. “Kitty. A beautiful girl, and false as
hell.”
“You wrong her,” I repeated. “You wrong these poor bones. Will
you listen?”
“Go on.” He steadied himself. “They won’t speak. Can you?”
“I’ll speak for them,” I continued. “In 1867 a government wagon
train was en route from Leavenworth for old Fort Bridger of Utah.”
“Very likely,” he sneered.
“There was a young wife with it, to join her husband at the post.
And there was a train attaché named Pierre Lavelle, half Spanish
and half Indian—a handsome scoundrel.”
“I’ll take your word for that.”
“He coveted the girl. She was innocent—she had no notion. One
evening after supper he and she rode up into a narrow draw, here in
western Wyoming, to seek flowers. He roped her and gagged her
and left her while he returned to the camp, on one pretext or another.
He succeeded in fastening a note inside her tent: ‘Tell my husband
I’ve gone with a better man.’”
“I got the note,” nodded the old man grimly. “Well?”
“The note was a forgery and a lie,” said I.
He sneered again.
“How do you know?”?

“I know. This first night he rode with the woman tied to her saddle;
the second night he freed her. He didn’t fear pursuit, and the trail and
the train were fifty miles behind. It was a lowering evening, and a
wild land. He advanced upon her, she smiled as if she had yielded,
but when he reached for her she struck him across the mouth and
snatched his knife from his belt and defied him.”
“Indeed? And how do you happen to know that, sir?”
“Wait. This stopped him for a moment. She fell upon her knees
and prayed to God for help. He wiped his lips and laughed. Can you
imagine that little scene, Mr. Brown? She in white, as she was——”
“She always loved white. There you are right,” conceded the old
man.
“And disheveled and at bay; he in his buckskins and greasy black
Indian hair, his lips bloody and his teeth glistening; and all the
country around promising no succor for her?”
“My imagination is dead,” he said. “Yours seems much alive. Well,
go on, go on.”
“Lavelle wiped his lips and laughed. ‘There’s no God in this region,
my lady,’ he mocked. ‘There’s only you and me.’”
“God-forsaken, God-forsaken,” the old man muttered. “A land
God-forsaken it is, as I have been.”
“Is it?” I challenged. “Wait. She prayed, and these are her very
words: ‘God, lift me from this fiend’s hands, or give me strength to lift
myself.’ Lavelle taunted: ‘Why not call upon your husband? He’ll be
hot to know. I left him just enough word to make him curious.’ And he
told her of the note. She cried: ‘Oh! How I hate you! Some day he
shall know, and know the truth. I hope he kills you.’ ‘Not for you, he
won’t,’ Lavelle answered. ‘He won’t want you after you’ve lived in my
Sioux lodge for a while.’”
The old man’s hands had clenched. He gazed fixedly as if
witnessing the scene.
“At this,” I proceeded, “she saw something in the fellow’s eyes that
alarmed her. When he rushed her she dodged and lunged, and
snapped the knife blade close to the hilt, upon his belt buckle. Then
she ran, leaving a strip of her dress in his fingers. She ran for higher
ground—ran like a hunted rabbit; sprang across a fissure, and

gained the top of a butte—a flat butte or mesa. And he made after,
jeering, for he knew that she had trapped herself. The mesa top
ended abruptly. Further flight was barred. He came on slowly,
enjoying her plight.”
The old man rasped:
“You say all this. How do you know? Answer me that.”
“Wait,” I bade. “Then she again fell on her knees, panting like a
nun of old Panama facing a buccaneer. But suddenly she called out,
this time gladly, and flung up her two arms, to the sky. And Lavelle
saw that which frightened even him. The north was strangely black
and jagged; out of the black there issued a roaring, and a gigantic
spectacle speeding very swiftly. It might have been the thunder bird
of the Sioux, said Lavelle, or a winged canoe, or monstrous
devouring demon—and it might have been an avalanching cloud of
wind and rain. But to her it was as if God were riding in upon a
thunderbolt chariot, and she had reached up her two arms to be
taken into that driving shelter.”
“And this happened, you say; did it?” smiled my old man,
sarcastically. “And you happen to know!”
“It happened, and I happen to know,” said I. “Lavelle was stopped
short again. The Indian in him recoiled. Then his ruffian courage
surged back within him. Whether god or spirit, it should not have her.
So he threw his rifle to his shoulder, and just as the blackness
swooped roaring and whistling to envelop her he touched trigger.
Then he ran headlong, in retreat out of the way. The cloud
descended, it passed, the rush of air in its wake knocked him flat, the
terror and the rain and the hail and the thunder and lightning
plastered him, face to the ground, at the foot of the mesa.”
“The morning the sun rose clear. But Lavelle could not get atop that
mesa again. The cloudburst had sheered away the approaches, like
a hydraulic stream, and washed them down as mud and gravel. The
mesa rose rimrocked and precipitous, like a biscuit to an ant. He
hallooed and got no answer. One horse had broken its tether; he
rode the other to a near-by ridge and gazed across. He could see
the girl lying white and motionless. His hawk eyes told him that she

was dead. So, being a coward in heart, he made off at speed. He
quit the country altogether, changed his name, drifted down into
border Arizona, and was shot at a gambling table in Tombstone
some forty years ago. The girl, you see,” said I, “has been lying here
ever since, the half coin—that half coin of promise in her fingers,
waiting for you and your understanding.”
“But,” he cried fiercely, “you say so. You weave a story. How am I
to know? Where is your proof? Why should I believe? How does it
happen——?”
“Because,” I answered, “these bones and this half coin ‘happen’ to
be here; and you ‘happen’ to be my passenger; and we ‘happen’ to
land together upon this ‘God-forsaken’ spot. And my middle name,”
said I, “‘happens’ to be Lavelle, from the line of my grandfather who
in his private memoirs confessed to a great wrong.”
My old man plumped to his knees; he groped for the half coin. I left
him pressing it to his lips and babbling a name, and I went back to
the plane.
Transcriber’s Note: This story appeared in the February 1926
issue of Weird Tales magazine.

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