A tale of dihedral groups, SL(2), and beyond

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A tale of dihedral groups, SL(2), and beyond

Or: N⁰-matrices, my love Daniel Tubbenhauer

t • 1 s

· · · ·

Joint work with Marco Mackaay, Volodymyr Mazorchuk and Vanessa Miemietz

July 2018

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LetA(Γ) be the adjacency matrix of a finite, connected, loopless graphΓ. Let U_e+1(X) be the Chebyshev polynomial .

Classification problem (CP).Classify allΓsuch that U_e+1(A(Γ)) = 0.

fore= 2

fore= 4 ADE graphs fore+ 2 being(at most)the Coxeter number.

Type Am: ^• ^• ^• ^{· · ·} ^• ^• ^• fore=m−1

Type Dm: • • · · · • •

•

for e= 2m−4

Type E6:

• • • • •

•

fore= 10

Type E7:

• • • • • •

•

fore= 16

Type E8:

• • • • • • •

•

fore= 28

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A3= 1 3 2

• • • A(A3) =



0 0 1

0 0 1

1 1 0



 S_A3={2 cos(^π₄),0,2 cos(^3π₄)}

U3(X) = (X−2 cos(^π₄))X(X−2 cos(^3π₄))

fore= 2

fore= 4 Smith∼1969. The graphs solutions to (CP) are precisely

ADE graphs fore+ 2 being(at most)the Coxeter number.

Type Am: ^• ^• ^• ^{· · ·} ^• ^• ^• fore=m−1

Type Dm: • • · · · • •

•

for e= 2m−4

Type E6:

• • • • •

•

fore= 10

Type E7:

• • • • • •

•

fore= 16

Type E8:

• • • • • • •

•

fore= 28

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A3= 1 3 2

• • • A(A3) =



0 0 1

0 0 1

1 1 0



 S_A3={2 cos(^π₄),0,2 cos(^3π₄)}

D4= 1 4 2

3

• •

•

A(D4) =





0 0 0 1

1 1 1 0



 S_D4={2 cos(^π₆),0²,2 cos(^5π₆)}

U3(X) = (X−2 cos(^π₄))X(X−2 cos(^3π₄))

U5(X) = (X−2 cos(^π₆))(X−2 cos(^2π₆))X(X−2 cos(^4π₆))(X−2 cos(^5π₆))

fore= 2

fore= 4 ADE graphs fore+ 2 being(at most)the Coxeter number.

Type Am: ^• ^• ^• ^{· · ·} ^• ^• ^• fore=m−1

Type Dm: • • · · · • •

•

for e= 2m−4

Type E6:

• • • • •

•

fore= 10

Type E7:

• • • • • •

•

fore= 16

Type E8:

• • • • • • •

•

fore= 28

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A3= 1 3 2

• • • A(A3) =



0 0 1

0 0 1

1 1 0



 S_A3={2 cos(^π₄),0,2 cos(^3π₄)}

D4= 1 4 2

3

• •

•

A(D4) =





0 0 0 1

1 1 1 0



 S_D4={2 cos(^π₆),0²,2 cos(^5π₆)}

U3(X) = (X−2 cos(^π₄))X(X−2 cos(^3π₄))

U5(X) = (X−2 cos(^π₆))(X−2 cos(^2π₆))X(X−2 cos(^4π₆))(X−2 cos(^5π₆)) fore= 2

fore= 4

Smith∼1969. The graphs solutions to (CP) are precisely ADE graphs fore+ 2 being(at most)the Coxeter number.

Type Am: ^• ^• ^• ^{· · ·} ^• ^• ^• fore=m−1

Type Dm: • • · · · • •

•

for e= 2m−4

Type E6:

• • • • •

•

fore= 10

Type E7:

• • • • • •

•

fore= 16

Type E8:

• • • • • • •

•

fore= 28

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A3= 1 3 2

• • • A(A3) =



0 0 1

0 0 1

1 1 0



 S_A3={2 cos(^π₄),0,2 cos(^3π₄)}

D4= 1 4 2

3

• •

•

A(D4) =





0 0 0 1

1 1 1 0



 S_D4={2 cos(^π₆),0²,2 cos(^5π₆)}

U3(X) = (X−2 cos(^π₄))X(X−2 cos(^3π₄))

U5(X) = (X−2 cos(^π₆))(X−2 cos(^2π₆))X(X−2 cos(^4π₆))(X−2 cos(^5π₆))

fore= 4

Smith∼1969. The graphs solutions to (CP) are precisely ADE graphs fore+ 2 being(at most)the Coxeter number.

Type Am: ^• ^• ^• ^{· · ·} ^• ^• ^• fore=m−1

Type Dm: • • · · · • •

•

fore= 2m−4

Type E6:

• • • • •

•

for e= 10

Type E7:

• • • • • •

•

for e= 16

Type E8:

•

for e= 28

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1 Dihedral representation theory

A brief primer onN0-representation theory DihedralN0-representation theory

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

3 Towards modular representation theory SL(2)

...and beyond

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The main example today: dihedral groups

The dihedral groups are of Coxeter type I₂(e+ 2):

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

=t_e+2i, e.g.: W₄=hs,t|s²=t²= 1, tsts=w₀=ststi

Example. These are the symmetry groups of regulare+ 2-gons, e.g. fore= 2 the Coxeter complex is:

• ^H

H

H F F

F F

1

I should do the Hecke case, but I will keep it easy.

For the moment: Never mind! Highest cell.

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

I will explain in a few minutes what cells are. For the moment: Never mind!

Lowest cell. Highest cell.

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

w0

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

w0

I will explain in a few minutes what cells are.

For the moment: Never mind!

Lowest cell.

Highest cell. s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

w0

Lowest cell.

Highest cell.

s-cell. t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

w0

Lowest cell.

Highest cell.

s-cell.

t-cell.

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The main example today: dihedral groups

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .sts

| {z }

e+2

=w₀=. . .tst

| {z }

e+2

• ^H

H

H F F

F F

1 t s

ts

st tst sts

w0

Lowest cell.

Highest cell.

s-cell.

t-cell.

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Dihedral representation theory on one slide

One-dimensional modules. Mλs,λt, λs, λt∈C, θs7→λs, θt7→λt.

e≡0 mod 2 e6≡0 mod 2

M_0,0,M_2,0, M_0,2,M_2,2 M_0,0,M_2,2

Two-dimensional modules. M_z,z ∈C, θs7→(²_{0 0}^z), θt7→(^{0 0}_z₂).

M_z,z ∈V^±_e−{0} M_z,z ∈V^±_e V_e =roots(U_e+1(X)) andV_e^± theZ/2Z-orbits under z 7→ −z.

The Bott–Samelson (BS) generatorsθs=s+ 1, θt=t+ 1.

There is also a Kazhdan–Lusztig (KL) bases. Explicit formulas do not matter today.

Proposition (Lusztig?).

The list of one- and two-dimensionalWe+2-modules is a complete, irredundant list of simple modules.

I learned this construction from Mackaay in 2017. Example.

M0,0is the sign representation andM2,2is the trivial representation. In casee is odd, Ue+1(X) has a constant term, soM2,0,M0,2are not representations.

Example.

Mz forz being a root of the Chebyshev polynomial is a representation because the braid relation in terms of BS generators

involves the coefficients of the Chebyshev polynomial. Example.

These representations are indexed byZ/2Z-orbits of the Chebyshev roots:

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Dihedral representation theory on one slide

M_0,0,M_2,0, M_0,2,M_2,2 M_0,0,M_2,2

I learned this construction from Mackaay in 2017.

Example.

involves the coefficients of the Chebyshev polynomial.

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Dihedral representation theory on one slide

M_0,0,M_2,0, M_0,2,M_2,2 M_0,0,M_2,2

The Bott–Samelson (BS) generatorsθs=s+ 1, θt=t+ 1. There is also a Kazhdan–Lusztig (KL) bases. Explicit formulas do not matter today.

I learned this construction from Mackaay in 2017.

Example.

M0,0is the sign representation andM2,2is the trivial representation.

In casee is odd, Ue+1(X) has a constant term, soM2,0,M0,2are not representations.

Example.

involves the coefficients of the Chebyshev polynomial. Example.

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Dihedral representation theory on one slide

M_0,0,M_2,0, M_0,2,M_2,2 M_0,0,M_2,2

Example.

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Dihedral representation theory on one slide

M_0,0,M_2,0, M_0,2,M_2,2 M_0,0,M_2,2

The Bott–Samelson (BS) generatorsθs=s+ 1, θt=t+ 1. There is also a Kazhdan–Lusztig (KL) bases. Explicit formulas do not matter today.

Example.

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N

0

-algebras and their representations

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

AP-moduleMwith a basisB^M is called anN0-module if xm∈N0B^M (x∈B^P,m∈B^M).

These are N0-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 upstairs.

Group algebras of finite groups with basis given by group elements areN⁰-algebras. The regular representation is anN⁰-module.

Example.

The regular representation of group algebras decomposes overCinto simples. However, this decomposition is almost never anN⁰-equivalence.

Example.

Hecke algebras of (finite) Coxeter groups with their KL basis areN⁰-algebras. For the symmetric group a ^miracle happens: all simples areN⁰-modules.

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N

0

-algebras and their representations

Example.

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

The regular representation is anN⁰-module.

Example.

The regular representation of group algebras decomposes overCinto simples. However, this decomposition is almost never anN⁰-equivalence.

Example.

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N

0

-algebras and their representations

Example.

The regular representation of group algebras decomposes overCinto simples.

However, this decomposition is almost never anN⁰-equivalence.

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N

0

-algebras and their representations

Example.

The regular representation of group algebras decomposes overCinto simples.

However, this decomposition is almost never anN⁰-equivalence.

Example.

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

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

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Cells of N

0

-algebras and N

0

-modules

Kazhdan–Lusztig ∼1979. x≤Lyifxappears inzywith non-zero coefficient for somez∈B^P. x∼Lyif x≤Lyandy≤Lx.

∼Lpartitions Pinto cells L. Similarly for right R, two-sided cells J orN0-modules.

AnN0-module Mis 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.

Example. TransitiveN0-modules arise as decategorifications 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 basic building blocks ofN⁰-representation theory”. 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 (Lusztig ≤2003).

Hecke algebras for the dihedral group with KL basis have the following cells:

1

s ts sts tsts ststs

t st tst stst tstst w0

We will see the transitiveN⁰-modules in a second.

Left cells Right cells Two-sided cells

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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

“The basic building blocks ofN⁰-representation theory”.

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.

1

s ts sts tsts ststs

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

x1

x2

x3

x4 m1

m2

m3

m4

1

s ts sts tsts ststs

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

x1

x2

x3

x4 m1

m2

m3

m4

Example.

1

s ts sts tsts ststs

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Cells of N

0

-algebras and N

0

-modules

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.

1

s ts sts tsts ststs

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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 (Lusztig ≤2003).

1

s ts sts tsts ststs

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Cells of N

0

-algebras and N

0

-modules

Example.

1

s ts sts tsts ststs w

Left cells

Right cells Two-sided cells

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Cells of N

0

-algebras and N

0

-modules

Philosophy.

Example.

1

s ts sts tsts ststs w0

Left cells

Right cells

Two-sided cells

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Cells of N

0

-algebras and N

0

-modules

Example.

1

s ts sts tsts ststs w

Left cells Right cells

Two-sided cells

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N

0

-modules via graphs

Construct aW_∞-moduleMassociated to a bipartite graphΓ:

M=Ch1,2,3,4,5i

1 3 2 4 5

H F H

F

θs Ms=

2 0 1 0 0 0 2 1 1 1 0 0 0 0 0











, θt Mt=

0 0 0 0 0 0 0 0 0 0 1 1 2 0 0













The adjacency matrixA(Γ) ofΓis

A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0













These areWe+2-modules for somee

only ifA(Γ) is killed by the Chebyshev polynomial Ue+1(X). Morally speaking: These are constructed as the simples but with integral matrices having the Chebyshev-roots as eigenvalues.

It is not hard to see that the Chebyshev–braid-like relation can not hold otherwise.

Hence, by Smith’s (CP) and Lusztig: We get a representation ofWe+2

ifΓis a ADE Dynkin diagram fore+ 2 being the Coxeter number. That these areN⁰-modules follows from categorification.

‘Smaller solutions’ are neverN0-modules.

Classification.

Complete , irredundant ^list of transitiveN⁰-modules ofWe+2:

Apex 1 cell s – t cell w0 cell

N⁰-reps. M0,0 MADE+bicoleringfor e+ 2 = Cox. num. M2,2

I learned this from/with Kildetoft–Mackaay–Mazorchuk–Zimmermann∼2016.

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N

0

-modules via graphs

M=Ch1,2,3,4,5i

1 3 2 4 5

θs

action

H F H

F

2 0 1 0 0 0 2 1 1 1













0 0 0 0 0 0 0 0 0 0













A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0









Classification.

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N

0

-modules via graphs

M=Ch1,2,3,4,5i

1 3 2 4 5

θs

action

H F H

F

θs Ms=

2 0 1 0 0 0 2 1 1 1 0 0 0 0 0











, θt Mt=

0 0 0 0 0 0 0 0 0 0 1 1 2 0 0













A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0













Classification.

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N

0

-modules via graphs

M=Ch1,2,3,4,5i

1 3 2 4 5

θs

action

H F H

F

2 0 1 0 0 0 2 1 1 1













0 0 0 0 0 0 0 0 0 0













A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0









Classification.

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N

0

-modules via graphs

M=Ch1,2,3,4,5i

1 3 2 4 5

θs

action

H F H

F

θs Ms=

2 0 1 0 0 0 2 1 1 1 0 0 0 0 0











, θt Mt=

0 0 0 0 0 0 0 0 0 0 1 1 2 0 0













A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0













Classification.

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N

0

-modules via graphs

M=Ch1,2,3,4,5i

1 3 2 4 5

θs

action

H F H

F

2 0 1 0 0 0 2 1 1 1













0 0 0 0 0 0 0 0 0 0













A(Γ) =

0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0









Classification.