2-representation theory in a nutshell

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2-representation theory in a nutshell

Or: N⁰-matrices, my love Daniel Tubbenhauer

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1 Philosophy: “Categorifying” classical representation theory Some classical results

Some categorical results

2 Some details

A brief primer onN0-representation theory A brief primer on 2-representation theory

Daniel Tubbenhauer 2-representation theory in a nutshell October 2018 2 / 15

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Pioneers of representation theory

LetGbe a finite group.

Frobenius∼1895++, Burnside∼1900++. Representation theory is the ^useful?

study of linear group actions

M: G−→ Aut(V),

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

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

“M(g) = a matrix inAut(V)”

We want to have a categorical version of this!

“M(a) = a matrix inEnd(V)”

We want to have a categorical version of this.

I am going to explain what we can do at present.

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Pioneers of representation theory

Frobenius∼1895++, Burnside∼1900++. Representation theory is the ^useful?

study of linear group actions

M: G−→ Aut(V),

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

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Pioneers of representation theory

LetAbe a finite-dimensional algebra.

Noether∼1928++. Representation theory is the useful? study of algebra actions M:A−→ End(V),

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

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Pioneers of representation theory

LetAbe a finite-dimensional algebra.

Noether∼1928++. Representation theory is the useful? study of algebra actions M:A−→ End(V),

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

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The strategy

“Groups, as men, will be known by their actions.” – Guillermo Moreno The study of group actions is of fundamental importance in mathematics and related field. Sadly, it is also very hard.

Representation theory approach. The analogous linear problem of classifying G-modules has a satisfactory answer for many groups.

Problem involving a group action

G X

Philosophy. Turn problems into linear algebra.

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The strategy

G X

Problem involving a linear group action

k[G] kX

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The strategy

G X

k[G] kX

“Decomposition of the problem”

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The strategy

G X

k[G] kX

k[G] L V_i new

insights?

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Some theorems in classical representation theory

B AllG-modules are built out of simples.

B The character of a simpleG-module is an invariant.

B There is an injection

{simpleG-modules}/iso ,→

{conjugacy classes inG}, which is 1 : 1 in the semisimple case.

B All simples can be constructed intrinsically using the regularG-module.

The ^character only remembers the traces of the acting matrices.

“RegularG-module

=Gacting on itself.”

Find categorical versions of these facts.

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Some theorems in classical representation theory

“RegularG-module

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Some theorems in classical representation theory

“RegularG-module

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Pioneers of 2-representation theory

Chuang–Rouquier & many others ∼2004++. Higher representation theory is the useful? study of (certain) categorical actions, e.g.

M:G−→Aut(V),

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

Mazorchuk–Miemietz∼2014. All(suitable)2-modules are built out of 2-simples (“weak2-Jordan–H¨older filtration”).

“M(g) = a functor inAut(V)”

Plus some coherence conditions which I will not explain.

The three goals of2-representation theory. Improve the theory itself.

Discuss examples. Find applications.

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Pioneers of 2-representation theory

LetC be a finitary 2-category.

Chuang–Rouquier & many others ∼2004++. Higher representation theory is

the ^useful? study of actions of 2-categories:

M:C −→End(V),

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

Mazorchuk–Miemietz∼2014. All(suitable)2-modules are built out of 2-simples (“weak2-Jordan–H¨older filtration”).

“M(g) = a functor inAut(V)”

Plus some coherence conditions which I will not explain.

The three goals of2-representation theory.

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The strategy – part two

2-Representation theory approach. The higher structure might give new insights into known group actions.

G X

“lift”

Example (Khovanov–Seidel & others∼2000++). There is a whole zoo of categorical actions of braid groups

which are “easily” shown to be faithful.

This is a big open problem for most braid groups and their modules.

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The strategy – part two

G X

Problem involving a categorical group action

“lift”

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The strategy – part two

G X

into 2-simples

“lift”

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The strategy – part two

G X

“lift”

new insights?

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The strategy – part two

G X

into 2-simples

“lift”

new insights?

Example (Khovanov–Seidel & others∼2000++).

There is a whole zoo of categorical actions of braid groups which are “easily” shown to be faithful.

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“Lifting” classical representation theory

Note that we have a very particular notion what a “suitable” 2-module is. What characters were for Frobenius

are these matrices for us.

There are some technicalities.

These turned out to be very interesting,

since their importance is only visible via categorification.

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“Lifting” classical representation theory

B All(suitable)2-modules are built out of 2-simples.

Note that we have a very particular notion what a “suitable” 2-module is.

What characters were for Frobenius are these matrices for us.

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“Lifting” classical representation theory

B The decategorified actions (a.k.a. matrices) of the M(F)’s are invariants.

Note that we have a very particular notion what a “suitable” 2-module is.

What characters were for Frobenius are these matrices for us.

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“Lifting” classical representation theory

{2-simples ofC}/equi.

,→

{certain (co)algebra 1-morphisms}/“2-Morita equi.”, which is 1 : 1 in well-behaved cases.

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“Lifting” classical representation theory

{2-simples ofC}/equi.

,→

{certain (co)algebra 1-morphisms}/“2-Morita equi.”, which is 1 : 1 in well-behaved cases.

B There exists principal 2-modules lifting the regular module.

Even in well-behaved cases there are 2-simples which do not arise in this way.

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N

0

-algebras and their modules

An algebraPwith a basisB^P with 1∈B^Pis called aN0-algebra if xy∈N0B^P (x,y∈B^P).

AP-moduleMwith a basisB^Mis called aN0-module if xm∈N0B^M (x∈B^P,m∈B^M).

These areN0-equivalent if there is aN0-valued change of basis matrix.

Example. N0-algebras andN0-modules arise naturally as the decategorification of 2-categories and 2-modules, andN0-equivalence comes from 2-equivalence.

Example.

Group algebras of finite groups with basis given by group elements areN⁰-algebras. The regular module is aN⁰-module.

Example.

The regular module of a group algebra decomposes overCinto simples. However, this decomposition is almost never anN⁰-equivalence.

(I will come back to this in a second.) Example.

Hecke algebras of (finite) Coxeter groups with their Kazhdan–Lusztig (KL) basis areN⁰-algebras.

For the symmetric group a ^miracle happens: all simples areN⁰-modules.

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N

0

-algebras and their modules

Example.

Group algebras of finite groups with basis given by group elements areN⁰-algebras.

The regular module is aN⁰-module.

Example.

The regular module of a group algebra decomposes overCinto simples. However, this decomposition is almost never anN⁰-equivalence.

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N

0

-algebras and their modules

Example.

The regular module of a group algebra decomposes overCinto simples.

However, this decomposition is almost never anN⁰-equivalence.

(I will come back to this in a second.)

Example.

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N

0

-algebras and their modules

Example.

The regular module of a group algebra decomposes overCinto simples.

However, this decomposition is almost never anN⁰-equivalence.

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Cells of N

0

-algebras and N

0

-modules

Clifford, Munn, Ponizovski˘ı∼1942++, Kazhdan–Lusztig ∼1979. x≤^Lyifx appears inzywith non-zero coefficient for z∈B^P. x∼^Lyif x≤^Lyandy≤^Lx.

∼Lpartitions Pinto left cells L. Similarly for right R, two-sided cells J or N0-modules.

AN0-moduleMis transitive if all basis elements belong to the same∼L

equivalence class. An apex ofMis a maximal two-sided cell not killing it.

Fact. Each transitiveN0-module has a unique apex.

Hence, one can study them cell-wise.

Example. TransitiveN0-modules arise naturally as the decategorification of simple 2-modules.

Philosophy.

Imagine a graph whose vertices are thex’s or them’s. v1→v2ifv1appears inzv2.

cells = connected components transitive = one connected component

“The simples or atoms ofN⁰-representation theory”.

Question (N⁰-representation theory).Classify them! Example.

Group algebras with the group element basis have only one cell,G itself. TransitiveN⁰-modules areC[G/H] forH being a subgroup. The apex isG.

Example (Kazhdan–Lusztig∼1979). Hecke algebras for the symmetric group with KL basis

have ^cells coming from the Robinson–Schensted correspondence.

The transitiveN⁰-modules are the simples

with apex given by elements for the same shape of Young tableaux. Example.

TakeG=Z/3Z. ThenG has three conjugacy classes and three associated simples. These are given by specifying a third root of unity.

G has only one non-trivial subgroup;G itself. The associatedN⁰-module is the regularG-module.

Moral.N⁰-representation theory studies modules which make ^sense in any characteristic.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

Imagine a graph whose vertices are thex’s or them’s.

v1→v2ifv1appears inzv2.

x1

x2

x3

x4 m1

m2

m3

m4

Example.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

x1

x2

x3

x4 m1

m2

m3

m4

Question (N⁰-representation theory).Classify them!

Example.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

x1

x2

x3

x4 m1

m2

m3

m4

Example.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

Example.

Group algebras with the group element basis have only one cell,G itself.

TransitiveN⁰-modules areC[G/H] forH being a subgroup. The apex isG.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

Example.

Example (Kazhdan–Lusztig∼1979).

Hecke algebras for the symmetric group with KL basis

with apex given by elements for the same shape of Young tableaux.

Example.

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

Example.

Example (Kazhdan–Lusztig∼1979).

Hecke algebras for the symmetric group with KL basis

with apex given by elements for the same shape of Young tableaux.

Example.

TakeG=Z/3Z. ThenG has three conjugacy classes and three associated simples.

These are given by specifying a third root of unity.

G has only one non-trivial subgroup;G itself.

The associatedN⁰-module is the regularG-module.

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Cell-modules

Natural, and computable, examples of transitiveN0-modules are the so-called cell modules which, in some sense, play the role of regular modules.

Fix a left cell L. LetM(≥L), respectivelyM(>_L), be the N0-modules spanned by allx∈B^P in the union L⁰≥LL, respectively L⁰>_LL.

We callC_L=M(≥L)/M(>_L) the (left) cell module for L.

Fact. “Cell⇒transitiveN0-module”.

Empirical fact. In well-behaved cases “Cell⇔transitiveN0-module”, and classification of transitiveN0-modules is fairly easy.

Question. Are there natural examples where “Cell6⇐ transitiveN0-module”?

Example. Decategorifications of cell 2-modules are key examples of cell modules.

Example.

C[G] with the group element basis has only one cell module, the regular module. However, the transitiveN⁰-modulesC[G/H] are cell modules forG/H.

So morally, “Cell⇔transitiveN⁰-module”. Example (Kazhdan–Lusztig∼1979, Lusztig∼1983++).

For Hecke algebras of the symmetric group with KL basis

“Cell⇔transitiveN⁰-module”. ^Example .

In general, for Hecke algebras the cell modules are Lusztig’s cell modules studied in connection with reductive groups in characteristicp.

Example.

Morally speaking, the more complicated the cell structure, the more likely that “Cell6⇐transitiveN⁰-module”.

Example

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Cell-modules

Example.

C[G] with the group element basis has only one cell module, the regular module.

However, the transitiveN⁰-modulesC[G/H] are cell modules forG/H.

So morally, “Cell⇔transitiveN⁰-module”.

Example (Kazhdan–Lusztig∼1979, Lusztig∼1983++). For Hecke algebras of the symmetric group with KL basis

Example.

Example

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Cell-modules

Example.

Example (Kazhdan–Lusztig∼1979, Lusztig∼1983++).

Example.

Example

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Cell-modules

Example.

Example (Kazhdan–Lusztig∼1979, Lusztig∼1983++).

Example.

Example

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2-representation theory in a nutshell

2-category categories functors nat. trafos

1-category vector spaces linear maps

0-category numbers

relate relate

relate categorify

categorify

categorify forms

forms

forms categorifies

categorifies

The ladder of categorification: in each step there is a new layer of structure which is invisible on the ladder rung below.

Goal.

Categorify the theory “representation theory” itself.

Observation.

A groupGcan be viewed as a single-object categoryG, and a module as a functor fromG

into the single-object categoryAut(V), i.e. M:G −→ Aut(V).

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2-representation theory in a nutshell

0-category numbers

relate relate

relate categorify

categorify

categorify forms

forms

forms categorifies

categorifies

Goal.

Observation.

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2-representation theory in a nutshell

0-category numbers

relate relate

relate categorify

categorify

categorify forms

forms

forms categorifies

categorifies

Goal.

Observation.

into the single-object categoryAut(V), i.e.

M:G −→ Aut(V).

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2-representation theory in a nutshell

0-category numbers

relate relate

relate categorify

categorify

categorify forms

forms

forms categorifies

categorifies

Classical representation theory lives here

Goal.

Observation.

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2-representation theory in a nutshell

0-category numbers

relate relate

relate categorify

categorify

categorify forms

forms

forms categorifies

categorifies

Classical representation theory lives here

2-representation theory should live here

Goal.

Observation.

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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

Goal.

Observation.

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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

categorifies

categorical module

Goal.

Observation.

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“Lifting” N

0

-representation theory

An additive,k-linear, idempotent complete, Krull–Schmidt 2-categoryC is called finitary if some finiteness conditions hold.

A simple transitive 2-module (2-simple) ofC is an additive,k-linear 2-functor M:C →A^f(= 2-cat of finitary cats),

such that there are no non-zero properC-stable ideals.

There is also the notion of 2-equivalence.

Mazorchuk–Miemietz∼2014.

2-Simples!simples (e.g. weak2-Jordan–H¨older filtration), but their decategorifications are transitiveN⁰-modules and usually not simple.

Define cell theory similarly as forN⁰-algebras and -modules. 2-simple⇒transitive, and transitive 2-modules have a 2-simple quotient.

Chan–Mazorchuk ∼2016.

Every 2-simple has an associated apex not killing it. Thus, we can again study them separately for different cells.

Example.

B-Mod(+fc=some finiteness condition)is a prototypical object ofA^f. A 2-module usually is given by endofunctors onB-Mod.

Example.

G can be (naively) categorified usingG-graded vector spacesVecG ∈A^f. The 2-simples are indexed by subgroupsHandφ∈H^∗(H,C^∗).

Example (Mazorchuk–Miemietz & Chuang–Rouquier & Khovanov–Lauda & ...).

2-Kac–Moody algebras(+fc)are finitary 2-categories. Their 2-simples are categorifications of the simples.

Example (Mazorchuk–Miemietz & Soergel & Khovanov–Mazorchuk–Stroppel & ...). Soergel bimodules for finite Coxeter groups are finitary 2-categories.

(Coxeter=Weyl: “Indecomposable projective functors onO⁰.”) Symmetric group: the 2-simples are categorifications of the simples. Example (Kildetoft–Mackaay–Mazorchuk–Miemietz–Zhang & ...).

Quotients of Soergel bimodules(+fc), e.g. small quotients, are finitary 2-categories. Except for the small quotients+the classification is widely open.

Example.

Fusion or modular categories are semisimple examples

of finitary 2-categories. (Think: Rep(G) or module categories of quantum groups.) Their 2-modules play a prominent role in quantum algebra and topology.

Question (“2-representation theory”). Classify all 2-simples of a fixed finitary 2-category.

This is the categorification of

‘Classify all simples a fixed finite-dimensional algebra’, but much harder, e.g. it is unknown whether there are always only finitely many 2-simples (probably not).

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“Lifting” N

0

-representation theory

Define cell theory similarly as forN⁰-algebras and -modules. 2-simple⇒transitive, and transitive 2-modules have a 2-simple quotient.

Example.

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“Lifting” N

0

-representation theory

Define cell theory similarly as forN⁰-algebras and -modules.

2-simple⇒transitive, and transitive 2-modules have a 2-simple quotient.

Example.

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“Lifting” N

0

-representation theory

Define cell theory similarly as forN⁰-algebras and -modules.

2-simple⇒transitive, and transitive 2-modules have a 2-simple quotient.

Every 2-simple has an associated apex not killing it.

Thus, we can again study them separately for different cells.

Example.