2-representations of Soergel bimodules

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2-representations of Soergel bimodules

Or: Mind your groups Daniel Tubbenhauer

2 1

1 2

1

left cells

“left modules”

2 1

1 2

1

right cells

“right modules”

2 1

1 2

1

two-sided cells

“bimodules”

2 1

1 2

1

H-cells

“subalgebras”

Joint with Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang

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Clifford, Munn, Ponizovski˘ı, Green∼1942++. Finite semigroups or monoids.

Example. N,Aut({1,...,n}) =S_n⊂T_n=End({1,...,n}), groups, groupoids, categories, any·closed subsets of matrices, “everything” ^click,etc.

The cell orders and equivalences:

x≤Ly ⇔ ∃z: y=zx, x∼Ly⇔(x ≤Ly)∧(y ≤Lx), x≤^R y⇔ ∃z⁰:y =xz⁰, x∼^R y⇔(x ≤^R y)∧(y ≤^R x), x≤LR y⇔ ∃z,z⁰:y =zxz⁰, x∼LR y ⇔(x ≤LR y)∧(y ≤LR x).

Left, right and two-sided cells: Equivalence classes.

Example (group-like). The unit 1 is always in the lowest cell –e.g. 1≤Ly because we can takez=y. Invertible elementsg are always in the lowest cell –i.e.

g ≤Ly because we can takez =yg⁻¹.

Theorem. (Mind your groups!) There is a one-to-one correspondence (simples with

apexJ(e) )

one-to-one

←−−−−→

(simples of (any) H(e)⊂ J(e)

) . Thus, the maximal subgroupsH(e) (semisimple overC) control

the whole representation theory (non-semisimple; even overC).

Example. (T3.)

H(e) =S3,S2,S1gives 3 + 2 + 1 = 6 associated simples. This is a general philosophy in representation theory. Buzz words. Idempotent truncations, Kazhdan–Lusztig cells,

quasi-hereditary algebras, cellular algebras,etc.

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Example (the transformation semigroupT₃). Cells – leftL(columns), right R (rows), two-sided J (big rectangles),H=L ∩ R(small rectangles).

(123),(213),(132) (231),(312),(321)

(122),(221) (133),(331) (233),(322)

(121),(212) (313),(131) (323),(232) (221),(112) (113),(311) (223),(332)

(111) (222) (333) Jlowest

J^middle Jbiggest

H ∼=S3

H ∼=S₂ H ∼=S1

Cute facts.

I EachHcontains precisely one idempotente or none idempotent. Eache is contained in someH(e). (Idempotent separation.)

I EachH(e) is a maximal subgroup. (Group-like.)

Theorem. (Mind your groups!) There is a one-to-one correspondence (simples with

apexJ(e) )

one-to-one

←−−−−→

Example. (T3.)

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(123),(213),(132) (231),(312),(321)

(122),(221) (133),(331) (233),(322)

(121),(212) (313),(131) (323),(232) (221),(112) (113),(311) (223),(332)

(111) (222) (333) Jlowest

J^middle Jbiggest

H ∼=S3

H ∼=S₂ H ∼=S1

Cute facts.

I Each simple has a unique maximalJ(e) whoseH(e) do not kill it. (Apex.) Theorem. (Mind your groups!)

There is a one-to-one correspondence (simples with

apexJ(e) )

one-to-one

←−−−−→

Example. (T3.)

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(123),(213),(132) (231),(312),(321)

(122),(221) (133),(331) (233),(322)

(121),(212) (313),(131) (323),(232) (221),(112) (113),(311) (223),(332)

(111) (222) (333) Jlowest

J^middle Jbiggest

H ∼=S3

H ∼=S₂ H ∼=S1

Cute facts.

I EachH(e) is a maximal subgroup. (Group-like.) Theorem. (Mind your groups!) There is a one-to-one correspondence (simples with

apexJ(e) )

one-to-one

←−−−−→

Example. (T3.)

H(e) =S3,S2,S1gives 3 + 2 + 1 = 6 associated simples.

This is a general philosophy in representation theory. Buzz words. Idempotent truncations, Kazhdan–Lusztig cells,

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(123),(213),(132) (231),(312),(321)

(122),(221) (133),(331) (233),(322)

(121),(212) (313),(131) (323),(232) (221),(112) (113),(311) (223),(332)

(111) (222) (333) Jlowest

J^middle Jbiggest

H ∼=S3

H ∼=S₂ H ∼=S1

Cute facts.

I Each simple has a unique maximalJ(e) whoseH(e) do not kill it. (Apex.) Theorem. (Mind your groups!)

There is a one-to-one correspondence (simples with

apexJ(e) )

one-to-one

←−−−−→

Example. (T3.)

H(e) =S3,S2,S1gives 3 + 2 + 1 = 6 associated simples.

This is a general philosophy in representation theory.

Buzz words. Idempotent truncations, Kazhdan–Lusztig cells, quasi-hereditary algebras, cellular algebras,etc.

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2-representation theory in a nutshell

2-moduleM i7→M(i)

category F7→M(F)

functor α7→ M(α)

nat. trafo

1-moduleM i7→ M(i)

vector space F7→M(F)

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-categories.

Monoidal categories, module categoriesRep(G) of finite groupsG, module categories of Hopf algebras, fusion or modular tensor categories, Soergel bimodulesS, categorified quantum groups, categorified Heisenberg algebras.

Examples of2-representations. 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 of2-representations.

Representation theory (classical and modular), link homology, combinatorics TQFTs, quantum physics, geometry.

Plan for today.

1) Give an overview of the main ideas of 2-representation theory. 2) Discuss the group-like exampleRep(G).

3) Discuss the semigroup-like exampleS. (Time flies: I will be brief.)

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-representations.

Categorical modules, functorial actions,

the LLT algorithm, cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module.

Applications of2-representations.

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Examples of2-representations.

Categorical modules, functorial actions,

the LLT algorithm, cyclotomic Hecke/KLR algebras, categorified (anti-)spherical module.

Applications of2-representations.

Plan for today.

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2-moduleM i7→M(i)

category F7→M(F)

nat. trafo

linear map

0-modulem i7→m(i)

number

categorical module

categorifies

Plan for today.

1) Give an overview of the main ideas of 2-representation theory.

2) Discuss the group-like exampleRep(G).

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Representation theory is group theory in vector spaces

LetCbe a finite-dimensional algebra.

Frobenius∼1895++, Burnside∼1900++, Noether∼1928++.

Representation theory is the ^useful? study of algebra actions M:C−→ End(V),

withVbeing some vector space. (Called modules or representations.) The “atoms” of such an action are called simple.

Maschke∼1899, Noether, Schreier∼1928. All modules are built out of simples (“Jordan–H¨older” filtration).

Basic question: Find the periodic table of simples.

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories); I will stick to this case for the rest of the talk,

but what I am going to explain works for 2-categories.

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2-representation theory is group theory in categories

LetC be a finitary 2-category.

Etingof–Ostrik, Chuang–Rouquier, Khovanov–Lauda, many others

∼2000++. 2-representation theory is the useful? study of actions of 2-categories:

M:C −→End(V),

withV being some finitary category. (Called 2-modules or 2-representations.) The “atoms” of such an action are called 2-simple (“simple transitive”).

Mazorchuk–Miemietz∼2014. All 2-modules are built out of 2-simples (“weak 2-Jordan–H¨older filtration”).

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories); I will stick to this case for the rest of the talk,

but what I am going to explain works for 2-categories.

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2-representation theory is group theory in categories

LetC be a finitary 2-category.

Etingof–Ostrik, Chuang–Rouquier, Khovanov–Lauda, many others

∼2000++. 2-representation theory is the useful? study of actions of 2-categories:

M:C −→End(V),

withV being some finitary category. (Called 2-modules or 2-representations.) The “atoms” of such an action are called 2-simple (“simple transitive”).

Mazorchuk–Miemietz∼2014. All 2-modules are built out of 2-simples (“weak 2-Jordan–H¨older filtration”).

Basic question: Find the periodic table of 2-simples.

Empirical fact.

Most of the fun happens already for monoidal categories (one-object 2-categories);

I will stick to this case for the rest of the talk, but what I am going to explain works for 2-categories.

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A category V is called finitary if its equivalent toC-pMod. In particular:

I It has finitely many indecomposable objectsM_j (up to∼=).

I It has finite-dimensional hom-spaces.

I Its Grothendieck group [V] = [V]_Z⊗ZCis finite-dimensional.

A finitary, monoidal category C can thus be seen as a categorification of a finite-dimensional algebra.

Its indecomposable objectsCi give a distinguished basis of [C].

A finitary 2-representation of C: I A choice of a finitary categoryV. I (Nice) endofunctorsM(Ci) acting onV. I [M(C_i)] giveN-matrices acting on [V].

The atoms (decat).

ACmodule is called simple if it has noC-stable ideals.

The atoms (cat).

AC 2-module is called 2-simple if it has noC-stable⊗-ideals.

Dictionary.

cat finitary finitary+monoidal fiat functors

decat vector space algebra self-injective matrices

Instead of studyingCand its action via matrices, studyC-pModand its action via functors.

Example (decat).

C=C= 1 acts on any vector space viaλ· . It has only one simpleV=C.

Example (cat).

C =Vec=Rep(1) acts on any finitary category viaC⊗C

It has only one 2-simpleV=Vec.

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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The atoms (decat).

The atoms (cat).

Dictionary.

Example (decat).

Example (cat).

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An algebra A= (A, µ, ι) inC:

µ=

A

A A

, ι=

1 A

, = , = = .

Its (right) modules (M, δ):

δ=

M

M A

, = , =.

Example. Algebras inVec are algebras; modules are modules.

Example. Algebras inRep(G) are discussed in a second.

The category of (right)A-modulesMod_C(AM) is a leftC 2-representation.

Theorem (spread over several papers). Completeness. For every 2-simpleM there exists

a simple(in the abelianization)algebra objectAM in(a quotient of)C (fiat) such thatM∼=Mod_C(AM).

Non-redundancy. M ∼=N if and only if A_M andA_N are Morita–Takeuchi equivalent.

Example.

Simple algebra objects inVecare simple algebras.

Up to Morita–Takeuchi equivalence these are justC; andMod_Vec(C)∼=Vec. The above theorem is a vast generalization of this.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M

M A

, = , =.

Theorem (spread over several papers). Completeness. For every 2-simpleM there exists

Example.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M

M A

, = , =.

Theorem (spread over several papers).

Completeness. For every 2-simpleM there exists

Example.

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µ=

A

A A

, ι=

1 A

, = , = = .

δ=

M

M A

, = , =.

Theorem (spread over several papers).

Completeness. For every 2-simpleM there exists

Example.

Up to Morita–Takeuchi equivalence these are justC; andMod_Vec(C)∼=Vec.

The above theorem is a vast generalization of this.

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Example (Rep(G)).

I LetC =Rep(G) (G a finite group).

I C is monoidal and finitary (and fiat). For anyM,N∈C, we haveM⊗N∈C: g(m⊗n) =gm⊗gn

for allg ∈G,m∈M,n∈N. There is a trivial representation1.

I The regular 2-representationM:C →End(C):

M //

f

M⊗

f⊗

N //N⊗

.

I The decategorification is aN-representation, the regular representation.

I The associated algebra object isA_M =1∈C.

Theorem (folklore?).

Completeness. All 2-simples ofRep(G) are of the formV(K, ψ). Non-redundancy. We haveV(K, ψ)∼=V(K⁰, ψ⁰)

⇔

the subgroups are conjugate andψ⁰=ψ^g, whereψ^g(k,l) =ψ(gkg⁻¹,glg⁻¹). Note thatRep(G) has only finitely many 2-simples.

This is no coincidence.

Theorem (Etingof–Nikshych–Ostrik ∼2004); the group-like case. IfC is fusion (fiat and semisimple),

then it has only finitely many 2-simples. This is false if one drops the semisimplicity. ^Example

Group-like; semisimple. There are not many interesting actions of groups on additive/abelian categories. Examples. Vec,Rep(G),Rep(Uq(g))^ss,

fusion or modular categoriesetc.

Semigroup-like; non-semisimple. There are many interesting actions of semigroups on additive/abelian categories.

Examples. Functors acting on categories, projective functors on categoryO, Soergel bimodules, categorified quantum groups and their Schur quotientsetc.

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Example (Rep(G)).

I LetK ⊂G be a subgroup.

I Rep(K) is a 2-representation of Rep(G), with action Res^G_K⊗ :Rep(G)→End(Rep(K)), which is indeed a 2-action because Res^G_K is a⊗-functor.

I The decategorifications areN-representations.

I The associated algebra object isA_M =Ind^G_K(1K)∈C.

⇔

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Example (Rep(G)).

I Letψ∈H²(K,C^∗). LetV(K, ψ) be the category of projectiveK-modules with Schur multiplierψ,i.e. vector spacesVwithρ: K → End(V) such that

ρ(g)ρ(h) =ψ(g,h)ρ(gh), for allg,h∈K. I Note thatV(K,1) =Rep(K) and

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

I V(K, ψ) is also a 2-representation ofC =Rep(G):

Rep(G) V(K, ψ) ^Res

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

I The decategorifications areN-representations. ^Example

I The associated algebra object isA^ψ_M =Ind^G_K(1K)∈C, but withψ-twisted multiplication.

⇔

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

Completeness. All 2-simples ofRep(G) are of the formV(K, ψ).

Non-redundancy. We haveV(K, ψ)∼=V(K⁰, ψ⁰)

⇔

the subgroups are conjugate andψ⁰=ψ^g, whereψ^g(k,l) =ψ(gkg⁻¹,glg⁻¹).

Note thatRep(G) has only finitely many 2-simples. This is no coincidence.

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

⇔

Note thatRep(G) has only finitely many 2-simples.

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

⇔

Note thatRep(G) has only finitely many 2-simples.

Theorem (Etingof–Nikshych–Ostrik∼2004); the group-like case.

IfC is fusion (fiat and semisimple), then it has only finitely many 2-simples.

This is false if one drops the semisimplicity. ^Example

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Example (Rep(G)).

⊗:V(K, φ)V(K, ψ)→ V(K, φψ).

G KId

−−−−−−→ Rep(K) V(K, ψ)−→ V^⊗ (K, ψ).

⇔

Group-like; semisimple.

There are not many interesting actions of groups on additive/abelian categories.

Examples. Vec,Rep(G),Rep(Uq(g))^ss, fusion or modular categoriesetc.

Semigroup-like; non-semisimple.

There are many interesting actions of semigroups on additive/abelian categories.

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Kazhdan–Lusztig ∼1979, Mazorchuk–Miemietz∼2010, many others.

Additive categories are like semigroups.

Example. BimA– the 2-category of projective bimodules over some

finite-dimensional algebra. Takee.g. Awith primitive idempotentse₁+e₂+e₃= 1, thenBim_A has ten indecomposable 1-morphismsA andAe_i⊗Ce_jA.

The cell orders and equivalences:

X≤LY⇔ ∃Z:Y⊂^⊕ZX, X∼LY⇔(X≤LY)∧(Y≤LX), X≤RY⇔ ∃Z⁰:Y⊂^⊕XZ⁰, X∼R Y⇔(X≤R Y)∧(Y≤R X), X≤LR Y⇔ ∃Z,Z⁰:Y⊂^⊕ZXZ⁰, X∼LR Y⇔(X≤LR Y)∧(Y≤LR X).

Left, right and two-sided cells: Equivalence classes.

Example (group-like). The monoidal unit 1is always in the lowest cell –e.g.

1≤Ly because we can takeZ=Y. Semisimple 1-morphisms Gwith dual are always in the lowest cell –i.e. G≤LYbecause we can takeZ=YG^∗.

Theorem (Mackaay–Mazorchuk–Miemietz–Zhang∼2017). IfC is fiat, then there is a one-to-one correspondence

(2-simples with apexJ

)

one-to-one

←−−−−→

(2-simples of (any) CH

) . CH is a certain 2-category supported onH.

Thus, theH-cells control the whole 2-representation theory.

Example. (BimA.)

H=Vectwice gives 1 + 1 = 2 associated 2-simples. Problem.

CHis rarely semisimple, left aside group-like. Counterexample. Taft category.

We need to work harder. Example (group-like).

Fusion categories,e.g. Rep(G), have only one cell.Rep(G)_H is everything. Example (semigroup-like).

LetRep(G,K) forKbeing of prime characteristic.

The projectives form a two-sided cell. Rep(G,K)_H can be complicated. Example (Kazhdan–Lusztig∼1979, Soergel∼1990).

Soergel bimodules S(Sn) for the symmetric group

have cells coming from the Robinson–Schensted correspondence. SH has one indecomposable object, but is ^not fusion.

Example (Taft algebraT2).

T2-Modhas two cells – the lowest cell containing the trivial representation; the biggest containing the projectives.

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Example (BimA for A as before). Cells – leftL(columns), rightR(rows), two-sidedJ (big rectangles),H=L ∩ R(small rectangles).

A

Ae₁⊗Ce₁A Ae₁⊗Ce₂A Ae₁⊗Ce₃A Ae2⊗Ce1A Ae2⊗Ce2A Ae2⊗Ce3A Ae₃⊗Ce₁A Ae₃⊗Ce₂A Ae₃⊗Ce₃A J^lowest

Jbiggest

H ∼=Vec H ∼=Vec

IfC is finitary, then each 2-simple has a unique maximalJ not killing it. (Apex.)

)

one-to-one

←−−−−→

Example. (BimA.)

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A

Jbiggest

H ∼=Vec H ∼=Vec

IfC is finitary, then each 2-simple has a unique maximalJ not killing it. (Apex.) Theorem (Mackaay–Mazorchuk–Miemietz–Zhang∼2017).

IfC is fiat, then there is a one-to-one correspondence (2-simples with

apexJ )

one-to-one

←−−−−→

Example. (BimA.)

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A

Jbiggest

H ∼=Vec H ∼=Vec

apexJ )

one-to-one

←−−−−→

Example. (BimA.)

H=Vectwice gives 1 + 1 = 2 associated 2-simples.

Problem. CHis rarely semisimple,

left aside group-like. Counterexample. Taft category.

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A

Jbiggest

H ∼=Vec H ∼=Vec

apexJ )

one-to-one

←−−−−→

Example. (BimA.)

H=Vectwice gives 1 + 1 = 2 associated 2-simples.

Problem.

CH is rarely semisimple, left aside group-like.

Counterexample. Taft category.

Example (group-like).

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A

Jbiggest

H ∼=Vec H ∼=Vec

)

one-to-one

←−−−−→

Example. (BimA.)

We need to work harder.

Fusion categories,e.g. Rep(G), have only one cell. Rep(G)_H is everything.

Example (semigroup-like).

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A

Jbiggest

H ∼=Vec H ∼=Vec

)

one-to-one

←−−−−→

Example. (BimA.)

The projectives form a two-sided cell. Rep(G,K)_H can be complicated.

Example (Kazhdan–Lusztig∼1979, Soergel∼1990).

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A

Jbiggest

H ∼=Vec H ∼=Vec

)

one-to-one

←−−−−→

Example. (BimA.)

have cells coming from the Robinson–Schensted correspondence.

SH has one indecomposable object, but is ^not fusion.

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A

Jbiggest

H ∼=Vec H ∼=Vec

)

one-to-one

←−−−−→

Example. (BimA.)

have cells coming from the Robinson–Schensted correspondence.

SH has one indecomposable object, but is ^not fusion.

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Categorify the H-cell theorem – Part II Theorem (Lusztig, Elias–Williamson ∼2012).

LetHbe anH-cell of W. There exists a fusion categoryAH such that:

I (1) For everyw ∈ H, there exists a simple objectA_w.

I (2) TheA_w, forw ∈ H, form a complete set of pairwise non-isomorphic simple objects.

I (3) The identity object isA_d, where d is the Duflo involution.

I (4)AH categorifiesA_H (think: the degree-zero part ofH_H) with [Aw] =aw

and

A_xA_y =L

z∈J γ^z_x,yA_z. vs. C_xC_y =L

z∈J v^a(z)h^z_x,yC_z+ bigger friends.

Here theγare the degree-zero coefficients of theh^z_x,y,i.e.

γ^z_x,y = (v^a(z)h^z_x,y)(0).

Examples in typeA1; coinvariant algebra.

C1=C[x]/(x²) andCs=C[x]/(x²)⊗C[x]/(x²). (Positively graded, but non-semisimple.) A1=CandAs=C⊗C. (Degree zero part.)

Takeaway messages.

(1) Group-like categories are easy, but slightly boring. (2) Semigroup-like categories are hard, but interesting.

(3) Try to reduce the semigroup-like case to the group-like case using Green’s theory. (4) This does not work in general use a positive grading.

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Categorify the H-cell theorem – Part II Theorem (Lusztig, Elias–Williamson ∼2012).

LetHbe anH-cell of W. There exists a fusion categoryAH such that:

I (1) For everyw ∈ H, there exists a simple objectA_w.

I (2) TheA_w, forw ∈ H, form a complete set of pairwise non-isomorphic simple objects.

I (3) The identity object isA_d, where d is the Duflo involution.

I (4)AH categorifiesA_H (think: the degree-zero part ofH_H) with [Aw] =aw

and

A_xA_y =L

z∈J γ^z_x,yA_z. vs. C_xC_y =L

z∈J v^a(z)h^z_x,yC_z+ bigger friends.

Here theγare the degree-zero coefficients of theh^z_x,y,i.e.

γ^z_x,y = (v^a(z)h^z_x,y)(0).

Takeaway messages.

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Categorify the H-cell theorem – Part II Theorem.

For any finite Coxeter groupW and anyH ⊂ J ofW, there is an injection Θ : {2-simples of AH}/∼=

,→ {graded 2-simples ofS with apexJ }/∼= I We conjecture Θ to be a bijection.

I We have proved (are about to prove) the conjecture for almost allH,e.g.

those containing the longest element of a parabolic subgroup ofW.

I If true, the conjecture implies that there are finitely many equivalence classes of 2-simples ofS.

I For almost allW, we would get a complete classification of the 2-simples.

Takeaway messages.