2-representations of Soergel bimodules—dihedral case

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2-representations of Soergel bimodules—dihedral case

Or: Who colored my Dynkin diagrams?

Daniel Tubbenhauer

t • 1 s

· · · ·

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

September 2019

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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 Ue+1(A(Γ)) = 0.

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

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

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

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

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

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

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: • • · · · • •

•

fore= 2m−4

Type E6:

• • • • •

•

for e= 10 Type E7:

• • • • • •

•

for e= 16 Type E8:

• • • • • • •

•

for e= 28

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1 A bit of motivation

2 Dihedral (2-)representation theory Classical vs. N-representation theory DihedralN-representation theory Categorified picture

3 Non-semisimple fusion rings The asymptotic limit

The limit v→0 of theN-representations Beyond

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gsemisimple Lie algebra givesO ⊃ O0.

Bernˇste˘ın–Gel’fand∼1980. Projective functorsP act onO0 and O0xP −−−→^decat. Z[W]x Z[W]

categorifies the regular representation of the associated Weyl groupW. Aside. Add grading and get Hecke algebra.

List of properties.

I P is additive, Krull–Schmidt, C-linear and monoidal, has finitely many indecomposables, and Hom-spaces are finite-dimensional. An adjoint of a projective functor is a projective functor. “Finitary/fiat acting 2-category”

I O ∼=A-pModforA a finite-dimensional algebra. “Finitary 2-module”

Question. What kind of theory governs such actions? Our answer. Finitary 2-representation theory.

Goal. Classify the “simplest” such actions. “Simple transitive 2-modules or 2-simples”

Example/Theorem (Bernˇste˘ın–Gel’fand∼1980).

g=slm.

2-simples are in 1:1 correspondence with simples ofW =Sm. Beyond this case not much was known.

Soergel∼1990++.

Soergel bimodulesS are a combinatorial, graded model ofP, and work for any Coxeter group.

Classifying 2-simples ofS is classifying 2-simples ofP.

The main theorem tomorrow will imply a complete classification of 2-simples forP for any semisimpleg, except two cases in typeE8.

Appearance of Soergel bimodules and there 2-representations in the wild. O, Hecke algebra, Kazhdan–Lusztig theory, braid group actions, link homologies, modular representation theory, 3-manifold invariants,

tensor and fusion categoriesetc. Today: Dihedral 2-representation theory. But keep in mind that we have a more general machinery

to study such questions. (More tomorrow.)

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List of properties.

g=slm.

Soergel∼1990++.

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List of properties.

g=slm.

Soergel∼1990++.

Soergel bimodulesSare a combinatorial, graded model ofP, and work for any Coxeter group.

Classifying 2-simples ofSis classifying 2-simples of P.

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List of properties.

g=slm.

Soergel∼1990++.

Soergel bimodulesSare a combinatorial, graded model ofP, and work for any Coxeter group.

Classifying 2-simples ofSis classifying 2-simples of P.

Appearance of Soergel bimodules and there 2-representations in the wild.

O, Hecke algebra, Kazhdan–Lusztig theory, braid group actions, link homologies, modular representation theory, 3-manifold invariants,

Today: Dihedral 2-representation theory. But keep in mind that we have a more general machinery

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List of properties.

g=slm.

Soergel∼1990++.

tensor and fusion categoriesetc.

Today: Dihedral 2-representation theory.

But keep in mind that we have a more general machinery to study such questions. (More tomorrow.)

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

W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

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:

Idea (Coxeter∼1934++).

Fact. The symmetries are given by exchanging flags.

Fix a flagF.

Fix a hyperplaneH0permuting the adjacent 0-cells ofF. Fix a hyperplaneH1permuting

the adjacent 1-cells ofF,etc. Write a vertexi for eachHi.

Connecti,jby ann-edge for Hi,Hjhaving angle cos(π/n).

This gives a generator-relation presentation.

And the braid relation measures the angle between hyperplanes.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

Example. These are the symmetry groups of regularIdea (Coxeter∼1934++).e+ 2-gons,e.g. fore= 2:

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• Idea (Coxeter∼1934++).

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

•

Fix a flagF.

Fix a hyperplaneH0permuting the adjacent 0-cells ofF.

Fix a hyperplaneH1permuting the adjacent 1-cells ofF,etc.

Write a vertexi for eachHi. Connecti,jby ann-edge for Hi,Hjhaving angle cos(π/n).

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

•

Fix a flagF.

the adjacent 1-cells ofF,etc.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• • •

Fix a flagF.

Write a vertexi for eachHi.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

•

cos(π/4)

• 4 •

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• 4

• •

1

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• 4

• •

1 t s

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• 4

• •

1 t s

ts

st

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• 4

• •

1 t s

ts

st tst sts

Fix a flagF.

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W_e+2=hs,t|s²=t²= 1, s_e+2 =. . .| {z }sts

e+2

=w₀=. . .| {z }tst

e+2

• 4

• •

1 t s

ts

st tst stsw0

Fix a flagF.

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Dihedral representation theory on one slide.

One-dimensional modules. Mλs,λt, λs, λt∈C,bs7→λs,bt7→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. Mz,z∈C,bs7→(²_{0 0}^z),bt7→(^{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 z7→ −z.

The Bott–Samelson (BS) generatorsbs=s+ 1,bt=t+ 1.

There is also a Kazhdan–Lusztig (KL) basiscw. We will nail it down later.

Proposition (Lusztig?).

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

I learned this construction 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.

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

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

The Bott–Samelson (BS) generatorsbs=s+ 1,bt=t+ 1. There is also a Kazhdan–Lusztig (KL) basiscw. We will nail it down later.

I learned this construction in 2017.

Example.

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

I learned this construction 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.

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

I learned this construction in 2017.Example.

Example.

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An algebraAwith afixed basisBÂ is called a (multi)N-algebra if xy∈NBÂ (x,y∈BÂ).

AA-moduleMwith afixedbasisB^Mis called aN-module if xm∈NB^M (x∈B^A,m∈B^M).

These areN-equivalent if there is aN-valued change of basis matrix.

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

Example (group like).

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

Fusion rings are with basis given by classes of simples areN-algebras. Key example: K0(Rep(G,C)) (easyN-representation theory). Key example: K0(Rep^ss_q(Uq(g)) =Gq) (intricateN-representation theory).

Example (semigroup like).

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

TheirN-representation theory is non-semisimple.

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

The regular module is anN-module.

Fusion rings are with basis given by classes of simples areN-algebras. Key example: K0(Rep(G,C)) (easyN-representation theory). Key example: K0(Rep^ss_q(Uq(g)) =Gq) (intricateN-representation theory).

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Fusion rings are with basis given by classes of simples areN-algebras.

Key example: K0(Rep(G,C)) (easyN-representation theory).

Key example: K0(Rep^ss_q(Uq(g)) =Gq) (intricateN-representation theory).

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Fusion rings are with basis given by classes of simples areN-algebras.

Key example: K0(Rep(G,C)) (easyN-representation theory).

Key example: K0(Rep^ss_q(Uq(g)) =Gq) (intricateN-representation theory).

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Clifford, Munn, Ponizovski˘ı, Green∼1942++, Kazhdan–Lusztig ∼1979.

x≤Lyifyappears inzxwith non-zero coefficient forz∈B^A. x∼Lyifx≤Ly andy≤Lx.

∼Lpartitions Ainto left cells L. Similarly for right R, two-sided cells LR or N-modules.

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

equivalence class. AnapexofMis a maximal two-sided cell not killing it.

Fact. Each transitiveN-module has a unique apex.

Hence, one can study them cell-wise.

Example. TransitiveN-modules arise naturally as the decategorification of 2-simples.

Group algebras with the group element basis have only one cell,G itself. TransitiveN-modules areC[G/H] forH⊂G subgroup/conjugacy. The apex is G.

Fusion rings in general have only one cell since each basis element [Vi] has a dual [Vi^∗] such that [Vi][Vi^∗] contains 1 as a summand.

Cell theory is useless for them! Example (Lusztig≤2003; semigroup like).

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 transitiveN-modules in a second.

Left cells. Right cells.

Two-sided cells.

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

TransitiveN-modules areC[G/H] forH⊂G subgroup/conjugacy. The apex is G.

Cell theory is useless for them! Example (Lusztig≤2003; semigroup like).

1

s ts sts tsts ststs

Two-sided cells.

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Cell theory is useless for them!

Example (Lusztig≤2003; semigroup like).

1

s ts sts tsts ststs

Two-sided cells.

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1

s ts sts tsts ststs w0

Left cells.

Right cells. Two-sided cells.

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1

Left cells.

Right cells.

Two-sided cells.

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1

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N-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

bs Ms=

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









, bt 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 like 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 areN-modules ^follows from categorification.

‘Smaller solutions’ are neverN-modules. Classification.

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

apex 1 cell s – t cell w0 cell

N-reps. M0,0 MADE+bicolering fore+ 2 = Cox. num. M2,2

I learned this from Kildetoft–Mackaay–Mazorchuk–Zimmermann∼2016. Fun fact about associated simples: ^Click .

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

1 3 2 4 5

bs

action

H F H

F

bs Ms=

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









, bt 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













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

1 3 2 4 5

bs

action

H F H

F

bs Ms=

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









, bt 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













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

1 3 2 4 5

bs

action

H F H

F

bs Ms=

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









, bt 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













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

1 3 2 4 5

bs

action

H F H

F

bs Ms=

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









, bt 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













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

1 3 2 4 5

bs

action

H F H

F

bs Ms=

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









, bt 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











