Representation theory of monoids and monoidal categories

Representation theory of monoids and monoidal categories
Or: Cells and actions
Daniel Tubbenhauer
Joint with Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang
June 2022
Daniel Tubbenhauer Representation theory of monoids and monoidal categories June 2022 1 / 7

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
Group reps Monoid reps
Fusion reps Fiat reps
generalize
categorify
rep theory
I Green, Clifford, Munn, Ponizovskiı̆ ∼1940+
++ many others
Representation theory of (finite) monoids
I Goal Find some categorical analog

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
Careful
There are zillions of choices involved
and I show you a categorification not the categorification
There is no unique way of doing this!
Actually, everything works for 2-categories/bicategories,
but I won’t touch this in this talk

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
Careful
There are zillions of choices involved
and I show you a categorification not the categorification
There is no unique way of doing this!
Actually, everything works for 2-categories/bicategories,
but I won’t touch this in this talk
Today
I explain monoid and fiat rep theory
fiat monoidal categories
categorify
−
−
−
−
−
→ certain fin dim algebras ⊃ monoid algebras
fiat reps
categorify
−
−
−
−
−
→ reps of certain fin dim algebras ⊃ monoid reps

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
Examples of monoidal categories
G-graded vector spaces V ectG , module categories Rep(G), same for monoids
Rep(Hopf algebra), tensor or fusion or modular categories,
Soergel bimodules (“the Hecke category”),
categorified quantum groups, categorified Heisenberg algebras, ...

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
Examples of reps of these
Categorical modules, functorial actions,
(co)algebra objects, conformal embeddings of affine Lie algebras, the LLT algorithm,
cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module, ...

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
Examples of reps of these
Categorical modules, functorial actions,
(co)algebra objects, conformal embeddings of affine Lie algebras, the LLT algorithm,
cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module, ...
Applications of categorical representations
Representation theory (classical and modular), link homologies, combinatorics,
TQFTs, quantum physics, geometry, ...

The theory of monoids (Green ∼1950+
+)
I Associativity ⇒ reasonable theory of matrix reps
I Southeast corner ⇒ reasonable theory of matrix reps

+)
Adjoining identities is “free” and there is no essential difference between
semigroups and monoids, or inverses semigroups and groups
The main difference is semigroups/monoids vs. inverses semigroups/groups
Today I will stick with the more familiar monoids and groups

+)
In a monoid information is destroyed
The point of monoid theory is to keep track of information loss

+)
Monoids appear naturally in categorification

+)
Example
Z is a group Integers
N is a monoid Natural numbers
Example
Cn = ha|an
= 1i is a group Cyclic group
Cn,p = ha|an+p
= ap
i is a monoid Cyclic monoid
Example
Sn = Aut({1, ..., n}) is a group Symmetric group
Tn = End({1, ..., n}) is a monoid Transformation monoid

+)
Finite groups are kind of random...

+)
The cell orders and equivalences:
Left, right and two-sided cells (a.k.a. L, R and J cells): equivalence classes.
I H cells = intersections of left and right cells
I Slogan cells measure information loss

+)

+)
I Each H contains no or 1 idempotent e; every e is contained in some H(e)
I Each H(e) is a maximal subgroup No internal information loss

+)
Example (group-like)
All invertible elements form the smallest cell

+)
Example (cells of N)
Every element is in its own cell, only 0 is idempotent

+)
Example (cells of C3,2, idempotent cells colored)

+)
Example (cells of T3, idempotent cells colored)

+)
Computing these “egg box diagrams” is one of the main tasks of monoid theory
GAP can do these calculations for you (package semigroups)

+)
Examples (no specific monoids)
Grey boxes are idempotent H cells

The simple reps of monoids
φ: S → GL(V ) S-representation on a K-vector space V , S is some monoid
I A K-linear subspace W ⊂ V is S-invariant if S W ⊂ W Substructure
I V 6= 0 is called simple if 0, V are the only S-invariant subspaces Elements
I Careful with different names in the literature: S-invariant !
subrepresentation, simple ! irreducible
I A crucial goal of representation theory
Find the periodic table of simple S-representations
S3

S3
Frobenius ∼1895+
+and others
For groups and K = C this theory is really satisfying

S3
Frobenius ∼1895+
+and others
For groups and K = C this theory is really satisfying
What about monoids?

Clifford, Munn, Ponizovskiı̆ ∼1940+
+(H-reduction)
There is a one-to-one correspondence

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Reps of monoids are controlled by their maximal subgroups
I Each simple has a unique maximal J (e) whose H(e) does not kill it Apex
I In other words (smod means the category of simples):
S-smodJ (e) ' H(e)-smod

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Example (groups)
Groups have only one cell – the group itself
H-reduction is trivial for groups

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Three simple reps over C:
one for 1 and two for Z/2Z

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Three simple reps over C:
one for 1 and two for Z/2Z
Six simple reps over C:
three for S3, two for S2 and one for S1

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Trivial rep of 1 induces to C3,2 and has apex Jb
Ja, Ja2 , Jt act by zero
Trivial rep of Z/2Z induces to C3,2 and has apex Jt
Nothing acts by zero

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Example (no specific monoid)
Five apexes: bottom cell, big cell, 2x2 cell, 3x3 cell, top cell
Simples for the 2x2 cell are acted on as zero by elements from 3x3 cell, top cell

+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)

Examples (no specific monoids)
All of these have four apexes

Categorification of monoid reps
I Usual answer ? = monoidal cats
I I need more structure than plain monoidal cats Specific categorification!

I Finitary = linear + additive + idempotent split + finitely many
indecomposables + fin dim hom spaces Cat of a fin dim algebra
I Fiat = finitary + involution + adjunctions + monoidal
I Fusion = fiat + semisimple
I Reps are on finitary cats
Finitary + fiat are additive analogs of tensor cats
Tensor cats as in Etingof–Gelaki–Nikshych–Ostrik ∼2015

Examples instead of formal defs
I Let C = Rep(G) (G a finite group)
I C is monoidal and nice. For any M, N ∈ C , we have M ⊗ N ∈ C :
g(m ⊗ n) = gm ⊗ gn
for all g ∈ G, m ∈ M, n ∈ N. There is a trivial representation 1
I The regular cat representation M : C → End(C ):
M //
f

M ⊗
f ⊗

N // N ⊗
I The decategorification is the regular representation

I Let K ⊂ G be a subgroup
I Rep(K) is a cat representation of Rep(G), with action
ResG
K ⊗ : Rep(G) → End(Rep(K)),
which is indeed a cat action because ResG
K is a ⊗-functor
I The decategorifications are N-representations

I Let ψ ∈ H2
(K, C∗
) (group field is now C)
I Let V(K, ψ) be the category of projective K-modules with Schur multiplier ψ,
i.e. vector spaces V with ρ: K → End(V) such that
ρ(g)ρ(h) = ψ(g, h)ρ(gh), for all g, h ∈ K
I Note that V(K, 1) = Rep(K) and
⊗: V(K, φ) V(K, ψ) → V(K, φψ)
I V(K, ψ) is also a cat representation of C = Rep(G):
Rep(G) V(K, ψ)
ResG
K Id
−
−
−
−
−
−
→ Rep(K) V(K, ψ)
⊗
−
→ V(K, ψ)

I Let ψ ∈ H2
(K, C∗
⊗: V(K, φ) V(K, ψ) → V(K, φψ)
Rep(G) V(K, ψ)
ResG
K Id
−
−
−
−
−
−
→ Rep(K) V(K, ψ)
⊗
−
→ V(K, ψ)
Classical
An S module is called simple (the “elements”)
if it has no S-stable ideals
We have the Jordan–Hölder theorem: every module is built from simples
Goal Find the periodic table of simples
Categorical
A C module is called simple (the “elements”)
if it has no C -stable monoidal ideals
We have the weak Jordan–Hölder theorem: every module is built from simples
Goal Find the periodic table of simples

I Let ψ ∈ H2
(K, C∗
⊗: V(K, φ) V(K, ψ) → V(K, φψ)
Rep(G) V(K, ψ)
ResG
K Id
−
−
−
−
−
−
→ Rep(K) V(K, ψ)
⊗
−
→ V(K, ψ)
Folk theorem?
Completeness All simples of Rep(G, C) are of the form V(K, ψ)
Non-redundancy We have V(K, ψ) ∼
= V(K0
, ψ0
)
⇔
the subgroups are conjugate and ψ0
= ψg
, where ψg
(k, l) = ψ(gkg−1
, glg−1
)

Clifford, Munn, Ponizovskiı̆ categorically
The cell orders and equivalences (X, Y, Z indecomposable, ⊂
⊕ = direct summand):
I H cells SH = Add(X ∈ H, 1) mod higher terms

Compare to monoids:
Indecomposables instead of elements, ⊂
⊕ instead of =
Otherwise the same!

Example (cells of Rep(G, C))
Only one cell since 1 ⊂
⊕ XX∗

⊕ XX∗
Example (cells of Rep(Z/3Z, F3), pseudo idempotent cells colored)

⊕ XX∗
Example (cells of the Hecke category of type B2, pseudo idempotent cells colored)

Categorical H-reduction

simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

Almost verbatim as for monoids
I Each simple has a unique maximal J whose SH does not kill it Apex
S -smodJ (e) ' SH-smod


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

No reduction


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

No reduction
Two apexes, three simples (2+1)


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

No reduction
Three apexes, four simples (1+2+1)

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
+)
+)
+)
+)
+)
+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

No reduction
There is still much to do...

Where are we?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
generalize
categorify
rep theory
++ many others
+)
+)
+)
+)
+)
+(H-reduction)

simples with
apex J (e)

one-to-one
←
−
−
−
−
→

simples of (any)
H(e) ⊂ J (e)


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ


simples with
apex J

one-to-one
←
−
−
−
−
→

simples of (any)
SH ⊂ SJ

No reduction
Thanks for your attention!

Representation theory of monoids and monoidal categories

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