What is...(2-)representation theory?

What is...(2-)representation theory?

Or: A (fairy) tale of matrices and functors Daniel Tubbenhauer

1 Classical representation theory Main ideas

Some classical results Some examples

2 Categorical representation theory Main ideas

Some categorical results An example

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

•

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1

1 0 0 1

,

1

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s

1 0 0 1

, 1 1

0 −1

,

1 s

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s t

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

1 s t

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s ts t

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

1 s t ts

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s ts t

st

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

1 s t ts st

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s ts t w0 st

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

A linearization of group theory

Slogan. Representation theory is group theory in vector spaces.

symmetries ofn-gons⊂ Aut(R²)

−e1

e1 e2

−e2 e1−e2 e2−e1

•

•

•

•

•

• 1 s ts t w0 st

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

These symmetry groups of the regularn-gons are the so-called dihedral groups D2n=hs,t|s²=t²= 1, . . .| {z }tsts

n

=w0=. . .| {z }stst

n

i which are the easiest examples of Coxeter groups.

Examplen= 4; its Coxeter complex.

•

•

•

•

• •

• •

1 t

s st ts tst

sts w0

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. A module is called semisimple if it is a direct sum of simples.

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.

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. A module is called semisimple if it is a direct sum of simples.

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.

Pioneers of representation theory

LetAbe a finite-dimensional algebra.

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

withVbeing some vector space. (Called modules or representations.)

The “atoms” of such an action are called simple. A module is called semisimple if it is a direct sum of simples.

Noether, Schreier∼1928. 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.

Pioneers of representation theory

LetAbe a finite-dimensional algebra.

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

withVbeing some vector space. (Called modules or representations.)

The “atoms” of such an action are called simple. A module is called semisimple if it is a direct sum of simples.

Noether, Schreier∼1928. 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.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices: 1 0

0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact. Semisimple case:

the character determines the module

!

mass determines the chemical compound. Example.

Z/2Z→ Aut(C²), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example. Z/2Z→ Aut(f2

2), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→ 1 1

0 1

and the module is non-simple, yet does not decompose. Morally: representation theory overZis never semisimple.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices:

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact. Semisimple case:

the character determines the module

!

mass determines the chemical compound. Example.

Z/2Z→ Aut(C²), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example. Z/2Z→ Aut(f2

2), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→ 1 1

0 1

and the module is non-simple, yet does not decompose. Morally: representation theory overZis never semisimple.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices:

1 0 0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact.

Semisimple case:

the character determines the module

!

mass determines the chemical compound.

Example. Z/2Z→ Aut(C²), 07→

1 0 0 1

& 17→ 0 1

1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example. Z/2Z→ Aut(f2

2), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→ 1 1

0 1

and the module is non-simple, yet does not decompose. Morally: representation theory overZis never semisimple.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices: 1 0

0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact. Semisimple case:

the character determines the module

!

mass determines the chemical compound.

Example.

Z/2Z→ Aut(C²), 07→

1 0 0 1

& 17→

0 1 1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example. Z/2Z→ Aut(f2

2), 07→ 1 0

0 1

& 17→ 0 1

1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→ 1 1

0 1

and the module is non-simple, yet does not decompose. Morally: representation theory overZis never semisimple.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices: 1 0

0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact. Semisimple case:

the character determines the module

!

mass determines the chemical compound.

Example.

Z/2Z→ Aut(C²), 07→

1 0 0 1

& 17→

0 1 1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example.

Z/2Z→ Aut(f2

2), 07→

1 0 0 1

& 17→

0 1 1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→

1 1 0 1

and the module is non-simple, yet does not decompose.

Morally: representation theory overZis never semisimple.

Life is non-semisimple

collection(“category”)of modules!the world modules!chemical compounds

simples!elements

semisimple!only trivial compounds non-semisimple!non-trivial compounds

Main goal of representation theory. Find the periodic table of simples.

Example.

Back to the dihedral group, an invariant of the module is the ^character χwhich only remembers the

traces of the acting matrices: 1 0

0 1

, 1 1

0 −1

,

−1 0

1 1

,

−1 −1

1 0

,

0 1

−1 −1

,

0 −1

−1 0

1 s t ts st sts=tst

w0

χ= 2 χ= 0 χ= 0 χ=−1 χ=−1 χ= 0

Fact. Semisimple case:

the character determines the module

!

mass determines the chemical compound.

Example.

Z/2Z→ Aut(C²), 07→

1 0 0 1

& 17→

0 1 1 0

Common eigenvectors: (1,1) and (1,−1) and base change gives 07→

1 0 0 1

& 17→

1 0 0 −1

and the module decomposes.

Example.

Z/2Z→ Aut(f2

2), 07→

1 0 0 1

& 17→

0 1 1 0

Common eigenvector: (1,1) and base change gives 07→

1 0 0 1

& 17→

1 1 0 1

and the module is non-simple, yet does not decompose.

Morally: representation theory overZis never semisimple.

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 analog 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.

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 analog linear problem of classifying G-modules has a satisfactory answer for many groups.

Problem involving a group action

G X

Problem involving a linear group action

k[G] kX

Philosophy. Turn problems into linear algebra.

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 analog linear problem of classifying G-modules has a satisfactory answer for many groups.

Problem involving a group action

G X

Problem involving a linear group action

k[G] kX

“Decomposition of the problem”

k[G] L V_i

Philosophy. Turn problems into linear algebra.

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 analog linear problem of classifying G-modules has a satisfactory answer for many groups.

Problem involving a group action

G X

Problem involving a linear group action

k[G] kX

“Decomposition of the problem”

k[G] L V_i new

insights?

Philosophy. Turn problems into linear algebra.

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.

“RegularG-module

=Gacting on itself.”

Find categorical versions of these facts.

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.

“RegularG-module

=Gacting on itself.”

Find categorical versions of these facts.

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.

“RegularG-module

=Gacting on itself.”

Find categorical versions of these facts.

Dihedral representation theory on one slide

One-dimensional modules. M^λs,λt, λs, λt∈C,s7→λs,t7→λt.

e≡0 mod 2 e6≡0 mod 2

M−1,−1,M1,−1,M−1,1,M1,1 M−1,−1,M1,1

Two-dimensional modules. Mz,z ∈C,s7→ ¹0−1^z

,t7→ ^{−1 0}z 1

.

n≡0 mod 2 n6≡0 mod 2

Mz,z ∈V(n)−{0} Mz,z ∈V(n) V(n) ={2 cos(πk/n−1)|k = 1, . . . ,n−2}.

Proposition (Lusztig?).

The list of one- and two-dimensionalD2n-modules is a complete, irredundant list of simples.

I learned this construction from Mackaay in 2017.

Note that this requires complex parameters. In particular, this does not work overZ.

Dihedral representation theory on one slide

One-dimensional modules. M^λs,λt, λs, λt∈C,s7→λs,t7→λt.

e≡0 mod 2 e6≡0 mod 2

M−1,−1,M1,−1,M−1,1,M1,1 M−1,−1,M1,1

Two-dimensional modules. Mz,z ∈C,s7→ ¹0−1^z

,t7→ ^{−1 0}z 1

.

n≡0 mod 2 n6≡0 mod 2

Mz,z ∈V(n)−{0} Mz,z ∈V(n) V(n) ={2 cos(πk/n−1)|k = 1, . . . ,n−2}.

Proposition (Lusztig?).

The list of one- and two-dimensionalD2n-modules is a complete, irredundant list of simples.

I learned this construction from Mackaay in 2017.

Note that this requires complex parameters. In particular, this does not work overZ.

Dihedral representation theory on one slide

One-dimensional modules. M^λs,λt, λs, λt∈C,s7→λs,t7→λt.

e≡0 mod 2 e6≡0 mod 2

M−1,−1,M1,−1,M−1,1,M1,1 M−1,−1,M1,1

Two-dimensional modules. Mz,z ∈C,s7→ ¹0−1^z

,t7→ ^{−1 0}z 1

.

n≡0 mod 2 n6≡0 mod 2

Mz,z ∈V(n)−{0} Mz,z ∈V(n) V(n) ={2 cos(πk/n−1)|k = 1, . . . ,n−2}.

Proposition (Lusztig?).

The list of one- and two-dimensionalD2n-modules is a complete, irredundant list of simples.

I learned this construction from Mackaay in 2017.

Note that this requires complex parameters.

In particular, this does not work overZ.

Beware of infinite dimensions

Take the infinite-dimensional Weyl algebraW=Chx, δ|δx= 1 +xδi. It has a very nice infinite-dimensional module

W→ End(C[X]), x7→ ·X, δ7→d/dX, andδx= 1 +xδjust becomes Leibniz’ product rule.

However, the classification of simples is not so easy. For example,Wdoes not have any finite-dimensional module.

Why? Assume it has andx 7→some matrixM; δ7→some matrixN. Then:

tr(MN) =tr(NM) = 1 +tr(MN) ⇒ 0 = 1.

But even there representation theory help

Take the infinite Artin–Tits groupB(C) =hb_i|. . .b_jb_ib_j

| {z }

mij

=b_ib_jb_i

| {z }

mij

i. ^Example

One can easily cook-up finite-dimensional modules which help to distinguish the elements ofB(C).

However, it is very hard and not known in general how to find faithful (“injective”) finite-dimensional modules.

Categorification in a nutshell

setN0

categoryVect

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

>

<

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

>

<

sur.

inj.

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

>

<

sur.

inj.

alln-dim.

vector spaces

V f W A universe itself!

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

>

<

sur.

inj.

alln-dim.

vector spaces

V f W A universe itself!

all rank 1 2−1-matrices

k² k g

A universe itself!

The point.

The categoryVecthas the whole power of linear algebra at hand! There is nothing comparable forN⁰:

N⁰is just a shadow ofVect.

Categorification in a nutshell

setN0

categoryVect

1 2

3 6

n k

k² k³

k⁶ kⁿ

dim.

+

⊕

·

⊗

>

<

sur.

inj.

alln-dim.

vector spaces

V f W A universe itself!

all rank 1 2−1-matrices

k² k g

A universe itself!

The point.

The categoryVecthas the whole power of linear algebra at hand!

There is nothing comparable forN⁰: N⁰is just a shadow ofVect.

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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

Classical representation theory lives here

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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

Classical representation theory lives here

2-representation theory should live here

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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

categorifies

categorifies

categorifies

categorifies

categorical module

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

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

categorifies

categorifies

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).

What one can hope for.

Problem involving a group action

G X

Problem involving a categorical group action

Decomposition of the problem into 2-simples

“lift”

new insights?

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

W→ End(C[X]) x7→ ·X δ7→d/dX

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

W→ End(L

i∈N0C{Xⁱ}) x7→L

i∈N0·X δ7→L

i∈N0d/dX

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

x7→L

i∈N0·X δ7→L

i∈N0d/dX W→End(L

i∈N0N_i-Mod)

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

δ7→L

i∈N0d/dX W→End(L

i∈N0N_i-Mod) x7→L

i∈N0Indⁱ⁺¹_i

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

W→End(L

i∈N0N_i-Mod) x7→L

i∈N0Indⁱ⁺¹_i δ7→L

i∈N0Resⁱ_i+1

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

The next ladder rung

Slogan. 2-representation theory is group theory in categories.

W=Chx, δ|δx= 1 +xδi

W→End(L

i∈N0N_i-Mod) x7→L

i∈N0Indⁱ⁺¹_i δ7→L

i∈N0Resⁱ_i+1

Step 1.

Replace the vector spacesC{Xⁱ}by appropriate categoriesNi-Mod.

HereNi are certain algebras(“Nil Coxeter”)which embed into each otherNi ,→Ni+1, of which we think about as liftingC{Xⁱ}−→^·X C{Xⁱ⁺¹}.

Step 2.

Replace the linear operators·X:C{Xⁱ} →C{Xⁱ⁺¹}by appropriate(“induction”)functorsIndⁱ⁺¹_i :Ni-Mod→Ni+1-Mod.

Step 3.

Replace the linear operatorsd/dX:C{Xⁱ⁺¹} →C{Xⁱ}by appropriate(“restriction”)functorsResⁱ_i+1:Ni-Mod→Ni+1-Mod.

Step 4.

Check that everything works.

In particular, the reciprocityResⁱ_i+1Indⁱ⁺¹_i ∼=Id⊕ Indⁱ⁻¹_i Resⁱ_i−1 categorifies Leibniz’ product rule.

Pioneers of 2-representation theory

LetGbe a finite group.

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.

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.

Improve the theory itself.

Discuss examples.

Find applications.

“Lifting” 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.

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.

“Lifting” classical representation theory

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

Goal 1. Improve the theory itself.

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.

“Lifting” classical representation theory

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

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

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.

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.

“Lifting” classical representation theory

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

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

B There is an injection

{2-simples ofC}/equi.

,→

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

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

Goal 1. Improve the theory itself.

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.

“Lifting” classical representation theory

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

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

B There is an injection

{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.

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.

2-modules of dihedral groups

Consider : θs=s+ 1, θt=t+ 1.

(Motivation. The Kazhdan–Lusztig basis has some neat integral properties.)

These elements generateC[D_2n] and their relations are fully understood:

θsθs= 2θs, θtθt= 2θt, a relation for . . .| {z }sts

n

=. . .| {z }tst

n

.

We want a categorical action. So we need:

B A categoryV to act on.

B EndofunctorsΘ_sandΘ_tacting onV.

B The relations ofθsandθthave to be satisfied by the functors.

B A coherent choice of natural transformations. (Skipped today.)

Some details.

Theorem∼2016.

There is a one-to-one correspondence {(non-trivial)2-simpleD2n-modules}/2-iso

←→1:1

{bicolored ADE Dynkin diagrams with Coxeter numbern}. Thus, its easy to write down a ^list .

Goal 2. Discuss examples.

2-modules of dihedral groups

Consider : θs=s+ 1, θt=t+ 1.

(Motivation. The Kazhdan–Lusztig basis has some neat integral properties.)

These elements generateC[D_2n] and their relations are fully understood:

θsθs= 2θs, θtθt= 2θt, a relation for . . .| {z }sts

n

=. . .| {z }tst

n

.

We want a categorical action. So we need:

B A categoryV to act on.

B EndofunctorsΘ_sandΘ_tacting onV.

B The relations ofθsandθthave to be satisfied by the functors.

B A coherent choice of natural transformations. (Skipped today.)

Some details.

Theorem∼2016.

There is a one-to-one correspondence {(non-trivial)2-simpleD2n-modules}/2-iso

←→1:1

{bicolored ADE Dynkin diagrams with Coxeter numbern}. Thus, its easy to write down a ^list .

Goal 2. Discuss examples.