sl 3 web algebras and categorified Howe duality Marco Mackaay - - PowerPoint PPT Presentation

sl3 web algebras and categorified Howe duality

Marco Mackaay (joint with Weiwei Pan and Daniel Tubbenhauer)

CAMGSD and University of the Algarve, Portugal

May 7, 2013 1

Some motivation

slk-Webs

Intertwiners

• Kauff, Kup, MOY
• Uq(slk)-Tensors

Resh-Tur

• Knot polynomials

2

Some motivation

slk-Webs

Intertwiners

• Kauff, Kup, MOY
• Uq(slk)-Tensors

Resh-Tur

• Knot polynomials

slk-Webs

Howe duality

• Kauff, Kup, MOY
• Uq(sln)-Irrep

Lusz, Caut, Kam, Lic, Morr

• Knot polynomials

3

Some motivation

slk-Foams/MF

Khov, Khov-Roz,M-S-V,...

• slk-Cycl string diags
• ???
• Webster
• Knot homologies

4

Some motivation

slk-Foams/MF

Khov, Khov-Roz,M-S-V,...

• slk-Cycl string diags
• ???
• Webster
• Knot homologies

slk-Foams/MF

Khov, Khov-Roz, M-S-V,...

• sln-Cycl KLR-algebra
• Howe 2-duality for k = 2, 3
• Chuang-Rouq, Lau-Quef-Rose
• Knot homologies

5

Skew Howe duality

The natural actions of GLk and GLn on Λp

are Howe dual (skew Howe duality). 6

Skew Howe duality

The natural actions of GLk and GLn on Λp

are Howe dual (skew Howe duality). This implies that InvSLk

• Λp1

⊗ Λp2

⊗ · · · ⊗ Λpn

∼ = W(p1, . . . , pn), where W(p1, . . . , pn) denotes the (p1, . . . , pn)-weight space of the GLn-module W(kℓ), if n = kℓ. 7

Plan

Let’s q-deform and categorify this for k = 3 (Cautis, Kamnitzer and Morrison did the q-deformation for arbitrary k ≥ 2) 8

Fundamental representation theory of Uq(sl3)

Let V+ be the basic Uq(sl3) representation and V− its dual V− := V ∗

+ ∼

= V+ ∧ V+. These are the two fundamental Uq(sl3) representations. 9

Fundamental representation theory of Uq(sl3)

Let V+ be the basic Uq(sl3) representation and V− its dual V− := V ∗

+ ∼

= V+ ∧ V+. These are the two fundamental Uq(sl3) representations. Let S = (s1, . . . , sn), with si ∈ {±}. We define VS = Vs1 ⊗ · · · ⊗ Vsn. 10

Generating intertwiners

∆++

−

: V− → V+ ⊗ V+ ∆−−

+

: V+ → V− ⊗ V− b−+ :

b+− :

d+− : V+ ⊗ V− →

d−+ : V− ⊗ V+ →

11

Kuperberg’s sl3-webs

Example + + − + − − Generating webs: ∆++

−

∆−−

+

b−+ b+− d+− d−+ 12

Let S = (s1, . . . , sn) be a sign string. Definition (Kuperberg) WS :=

where IS is generated by: = [3] (0.1) = [2] (0.2) = + (0.3) 13

Let S = (s1, . . . , sn) be a sign string. Definition (Kuperberg) WS :=

where IS is generated by: = [3] (0.1) = [2] (0.2) = + (0.3) From (0.1), (0.2) and (0.3) it follows that any w ∈ WS is a linear combination of non-elliptic webs (no circles, digons or squares). The latter form a basis, BS. 14

Kuperberg’s Theorem

Webs correspond to intertwiners Theorem (Kuperberg) WS ∼ = Hom( 1, VS) ∼ = Inv(VS). BS is called the web basis of Inv(VS). 15

The general and special linear quantum groups

Definition i) Uq(gln) is generated by K±1

1 , . . . , K±1 n , E±1, . . . , E±(n−1),

subject to (αi = εi − εi+1 = (0, . . . , 1, −1, . . . , 0) ∈

KiKj = KjKi KiK−1

i

= K−1

i

Ki = 1 EiE−j − E−jEi = δi,j KiK−1

i+1 − K−1 i

Ki+1 q − q−1 KiE±j = q±(εi,αj)E±jKi + some extra relations we won’t need today ii) Uq(sln) ⊆ Uq(gln) is generated by KiK−1

i+1 and E±i.

16

Idempotented quantum groups

Definition (Beilinson-Lusztig-MacPherson) For each λ ∈

1λ1µ = δλ,ν1λ E±i1λ = 1λ±αiE±i Ki1λ = qλi1λ. Define ˙ U(gln) =

• λ,µ∈Zn

1λUq(gln)1µ. Define ˙ U(sln) similarly by adjoining idempotents 1µ to Uq(sln) for µ ∈

17

Back to q-skew Howe duality

Definition An enhanced sign sequence is a sequence S = (s1, . . . , sn) with si ∈ {∅, −1, 1, ×}, for all i = 1, . . . n. The corresponding weight µ = µS ∈ Λ(n, d) is given by the rules µi =            if si = ∅ 1 if si = 1 2 if si = −1 3 if si = × . Let Λ(n, d)3 ⊂ Λ(n, d) be the subset of 3-bounded weights. 18

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. 19

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. We define WS := W

S,

and BS := B

S.

20

Let n = d = 3ℓ. For any enhanced sign string S, we define S by deleting the entries equal to ∅ or ×. We define WS := W

S,

and BS := B

S.

Definition Define W(3ℓ) :=

• µS∈Λ(n,n)3

WS. 21

The action

Define ϕ: ˙ U(gln) → End

• W(3ℓ)
• 1λ →

λ1 λ2 λn

E±i1λ →

• λ1

λi−1 λi λi+1 λi±1 λi+1∓1 λi+2 λn

Conventions: vertical edges labeled 1 are oriented upwards, vertical edges labeled 2 are oriented downwards and edges labeled 0 or 3 are erased. 22

Examples

E+11(22) →

2 2 3 1

E−2E+11(121) →

1 2 1 2 2

23

The isomorphism from q-skew Howe duality

Lemma The map ϕ gives rise to an isomorphism ϕ: V(3ℓ) → W(3ℓ)

• f ˙

U(gln)-modules. Note that the empty web wh := w(3ℓ), which generates W(×k,∅2k) ∼ =

24

The categorified story

Let’s categorify everything 25

sl3 Foams

Consider formal

singular cobordisms, e.g. the zip and unzip: 26

sl3 Foams

Consider formal

singular cobordisms, e.g. the zip and unzip: We also allow dots, which cannot cross singular arcs. 27

sl3 Foams

Consider formal

singular cobordisms, e.g. the zip and unzip: We also allow dots, which cannot cross singular arcs. Mod out by the ideal generated by ℓ = (3D, NC, S, Θ) and the closure relation: 28

Khovanov’s local relations: ℓ = (3D, NC, S, Θ)

= 0 (0.4) = − − − (0.5) = = 0, = −1 (0.6) =    1 (α, β, γ) = (1, 2, 0) or a cyclic permutation −1 (α, β, γ) = (2, 1, 0) or a cyclic permutation else (0.7) The relations in ℓ suffice to evaluate any closed foam! 29

The category of foams

Let Foam3 be the category of webs and foams. 30

The category of foams

Let Foam3 be the category of webs and foams. Other relations in Foam3 are: = − (Bamboo) = − (RD) 31

More relations in Foam3

= 0 (Bubble) = − (DR) = − − (SqR) 32

More relations in Foam3

+ + = + + = 0 = 0 (Dot Migration) 33

The grading

The q-grading of a foam U is defined as q(U) := χ(∂U) − 2χ(U) + 2d + b. This makes Foam3 into a graded category. 34

Foam homology

Definition The foam homology of a closed web w is defined by F(w) := Foam3(∅, w). 35

Foam homology

Definition The foam homology of a closed web w is defined by F(w) := Foam3(∅, w). F(w) is a graded complex vector space, whose q-dimension can be computed by the Kuperberg bracket:

1

• w ∐
• = [3]w

2

• = [2]
• 3
• =
• +
• The relations above correspond to the decomposition of F(w)

into direct summands. 36

Define w∗ by

w w*

(0.8) Define uv∗ by

u v*

(0.9) Define v∗u by

u v*

(0.10) 37

The web algebra

Definition (M-P-T) Let S = (s1, . . . , sn). The web algebra KS is defined by KS :=

• u,v∈BS

uKv,

with

uKv := F(u∗v){n}.

Multiplication is defined as follows:

uKv1 ⊗ v2Kw → uKw

is zero, if v1 = v2. If v1 = v2, use the multiplication foam, e.g. 38

The multiplication foam

v w* v v* w* v

39

The multiplication foam

v w* v v* w* v

Lemma (M-P-T) The multiplication foam mv only depends on the isotopy type of v and has q-degree n, so KS is a graded algebra. 40

Question: is KS unital and associative? 41

Question: is KS unital and associative? YES! For any u, v ∈ BS, we have a grading preserving isomorphism Foam3(u, v) ∼ = uKv. Using this isomorphism, the multiplication

uKv ⊗ v′Kw → uKw

corresponds to the composition Foam3(u, v) ⊗ Foam3(v′, w) → Foam3(u, w), if v = v′, and is zero otherwise. 42

Example

v* v v v*

Note that we have 1 =

• u∈BS

1u ∈ KS. 43

For any enhanced sign string, define KS := K

S.

44

For any enhanced sign string, define KS := K

S.

Define also K(3ℓ) :=

• µS∈Λ(n,n)3

KS and W(3ℓ) :=

• µS∈Λ(n,n)3

KS-pmodgr. 45

For any enhanced sign string, define KS := K

S.

Define also K(3ℓ) :=

• µS∈Λ(n,n)3

KS and W(3ℓ) :=

• µS∈Λ(n,n)3

KS-pmodgr. I will explain that categorified Howe duality implies Proposition (M-P-T) K

(W(3ℓ)) ∼ = W(3ℓ). 46

Khovanov and Lauda’s categorification of ˙ U(sln)

Definition (Khovanov-Lauda) The 2-category U(sln) consists of

• bjects: λ ∈

a 1-morphism from λ to λ′ is a formal finite direct sum of 1-morphisms Ei1λ{t} = 1λ′Ei1λ{t} for any t ∈

λ′ = λ + iX. The 2-morphisms are

• f:

47

The generating 2-morphisms

For any i, the identity 1Ei1λ{t} 2-morphism is represented as · · · i1 i2 im i1 i2 im λ λ + iX The strand labelled iα is oriented up if εα = + and oriented down if εα = −. 48

The generating 2-morphisms

For each λ ∈

• i
• λ

λ + iX

• i
• λ

λ + iX

• i

j

λ

• i

j

λ i · i i · i −i · j −i · j

• i

λ

• i

λ

• i

λ

• i

λ 1 + λi 1 − λi 1 + λi 1 − λi 49

The relations on 2-morphisms

The one color relations:

i) Planar isotopies ii) Nil-Hecke relations such as (recall E2

i = [2]E(2) i

)

• λ

i i

=

• i

i

λ −

• i

i

λ iii) All dotted bubbles of negative degree are zero and a dotted bubble of degree zero equals ±1 iv) (Recall EiFi1λ = FiEi1λ + [λi]1λ)

• i

i

λ λ =

• λ

i i

−

λi−1

• f=0

f

• g=0

λ

• f−g
• λi−1−f

i

• −λi−1+g

50

More color relations, e.g: for i = j (recall EiE−j = E−jEi)

• λ

i j

=

• λ

i j

• λ

i j

=

• λ

i j

51

More color relations, e.g: for i = j (recall EiE−j = E−jEi)

• λ

i j

=

• λ

i j

• λ

i j

=

• λ

i j

Theorem (Khovanov-Lauda) Let ˙ U(sln) be the Karoubi envelope of U(sln). Then K

( ˙ U(sln)) ∼ = ˙ U(sln). 52

More color relations, e.g: for i = j (recall EiE−j = E−jEi)

• λ

i j

=

• λ

i j

• λ

i j

=

• λ

i j

Theorem (Khovanov-Lauda) Let ˙ U(sln) be the Karoubi envelope of U(sln). Then K

( ˙ U(sln)) ∼ = ˙ U(sln). Definition U(gln) is obtained from U(sln) by switching to gln–weights. 53

The cyclotomic KLR algebras

Definition Let λ ∈ Λ(n, n)+. The finite-dimensional cyclotomic KLR-algebra Rλ is defined by taking all downward diagrams in U(sln)1λ and modding out by the ideal generated by

• i1 i2 i3

im λm:=λm−λm+1

λ 54

The cyclotomic KLR algebras

Definition Let λ ∈ Λ(n, n)+. The finite-dimensional cyclotomic KLR-algebra Rλ is defined by taking all downward diagrams in U(sln)1λ and modding out by the ideal generated by

• i1 i2 i3

im λm:=λm−λm+1

λ Definition Vλ := Rλ-pmodgr. 55

The cyclotomic quotient theorem

The following result was conjectured by Khovanov and Lauda in 2008: Theorem (Brundan-Kleshchev, Lauda-Vazirani, Webster, Kang-Kashiwara,...) There is a graded categorical action of U(sln) (resp. U(gln)) on Vλ and K

(Vλ) ∼ = Vλ as ˙ U(sln)-modules (resp. ˙ U(gln)-modules). 56

The categorical action

We define a categorical action of U(gln) (resp. U(sln)) on W(3ℓ). 57

The categorical action

We define a categorical action of U(gln) (resp. U(sln)) on W(3ℓ). On objects: use ϕ: ˙ U(gln) → End

• W(3ℓ)
• .

58

The categorical action

We define a categorical action of U(gln) (resp. U(sln)) on W(3ℓ). On objects: use ϕ: ˙ U(gln) → End

• W(3ℓ)
• .

On morphisms: we give a list of the foams associated to the generating morphisms of U(gln). Warning: facets labeled 0 or 3 have to be removed. 59

The categorical action on morphisms, e.g.

• i,λ

→

• λi

λi+1

• i,i,λ →

−

• λi

λi+1

• i,i+1,λ →

(−1)λi+1

λi λi+1 λi+2

60

The categorical action on morphisms, e.g.

• i,λ

→

λi λi+1

61

The categorical action on morphisms, e.g.

• i,λ

→

λi λi+1

Proposition (M-P-T) The categorical action of U(gln) (resp. U(sln)) on W(3ℓ) is degree preserving and well-defined. 62

Examples: categorical action preserves degrees

• 1,(12)

→ −

1 2 3 =: f

and deg(

1,(12)) = 2.

63

Examples: categorical action preserves degrees

• 1,(12)

→ −

1 2 3 =: f

and deg(

1,(12)) = 2.

χ(f) = 12 − 14 + 3 = 1 χ(∂f) = 12 − 12 = 0 b = 4 Total result: q(f) = 0 − 2 + 4 = 2 64

Example: the KL-relations are preserved

• i

i

(12) (12) =

• (12)

i i

−

• (12)

65

Example: the KL-relations are preserved

• i

i

(12) (12) =

• (12)

i i

−

• (12)

becomes the (SqR)-relation = − − . 66

Example: the KL-relations are preserved

• i

i

(12) (12) =

• (12)

i i

−

• (12)

becomes the (SqR)-relation = − − . Note that the signs match perfectly: sign

• 1,(12)
• = +

and sign

• 1,(12)
• = −

67

Harvest time

By Rouquier’s universality theorem, we get Theorem (M-P-T) The algebras R(3ℓ) and K(3ℓ) are Morita equivalent, i.e. there exists an equivalence of the U(sln) 2-representations Φ: R(3ℓ)-modgr → K(3ℓ)-modgr. This equivalence also restricts to an equivalence of the U(sln) 2-representations Φ: V(3ℓ) = R(3ℓ)-pmodgr → W(3ℓ) = K(3ℓ)-pmodgr. 68

Projective modules

For any u ∈ BS, define Pu :=

• v∈BS

vKu ∈ KS-pmodgr.

Remark (Khovanov-Kuperberg, Morrison-Nieh) In general, Pu is not indecomposable. 69

Projective modules

For any u ∈ BS, define Pu :=

• v∈BS

vKu ∈ KS-pmodgr.

Remark (Khovanov-Kuperberg, Morrison-Nieh) In general, Pu is not indecomposable. Define γS : WS → K

(WS) by γS(u) := [Pu]. Lemma (M-P-T) The map γS is injective. 70

Harvest time

Checking all the definitions, we get V(3ℓ)

γV

− − − − → K

(V(3ℓ))

ϕ

 

• K0(Φ)

 

• W(3ℓ)

γW

− − − − → K

(W(3ℓ)) . 71

Harvest time

Checking all the definitions, we get V(3ℓ)

γV

− − − − → K

(V(3ℓ))

ϕ

 

• K0(Φ)

 

• W(3ℓ)

γW

− − − − → K

(W(3ℓ)) . Corollary (M-P-T) The map γW is an isomorphism of ˙ U(sln)-modules. In particular, we have WS ∼ = K

(KS). 72

Harvest time

Proposition (M-P-T) For each u ∈ BS, there exists an indecomposable projective module Qu ∈ KS-pmodgr such that Pu ∼ = Qu ⊕

• v<u

c(u, v)Qv with c(u, v) ∈

73

Harvest time

Proposition (M-P-T) For each u ∈ BS, there exists an indecomposable projective module Qu ∈ KS-pmodgr such that Pu ∼ = Qu ⊕

• v<u

c(u, v)Qv with c(u, v) ∈

Theorem (M-P-T) The isomorphism γS : WS → K

(KS) maps the dual canonical basis of WS to {[Qu] | u ∈ BS}. 74

Harvest time

Let X(3ℓ)

µS

be the Spaltenstein variety of partial flags {0} = V0 ⊆ V1 ⊆ · · · ⊆ Vn =

such that dim Vi/Vi−1 = µi and x(3ℓ)Vi ⊆ Vi−1 with x(3ℓ) a fixed nilpotent matrix of Jordan type (ℓ3) = (3ℓ)T . Theorem (Brundan-Ostrik, Brundan-Kleshchev, M-P-T) We have Z(KS) ∼ = H∗(X(3ℓ)

µS ).

75

Further questions

Can KS be obtained using the intersection cohomology of the Fontaine-Kamnitzer-Kuperberg web varieties? 76

Further questions

Can KS be obtained using the intersection cohomology of the Fontaine-Kamnitzer-Kuperberg web varieties? KS is a graded cellular algebra (Mathas-Hu). Does it have a quasi-hereditary cover (categorifying the full VS)? Relation with Webster’s categorification of VS and Stroppel and Webster’s quiver Schur algebras? This is joint work in progress with Pan and Tubbenhauer. 77

Further questions

Can KS be obtained using the intersection cohomology of the Fontaine-Kamnitzer-Kuperberg web varieties? KS is a graded cellular algebra (Mathas-Hu). Does it have a quasi-hereditary cover (categorifying the full VS)? Relation with Webster’s categorification of VS and Stroppel and Webster’s quiver Schur algebras? This is joint work in progress with Pan and Tubbenhauer. What about k > 3? Yonezawa and I have categorified the slk web spaces using matrix factorizations and also defined a categorical action of U(sln) on these. Hopefully, we can prove the analogue for k > 3 of all the results I have shown you today. 78

Further questions

Can KS be obtained using the intersection cohomology of the Fontaine-Kamnitzer-Kuperberg web varieties? KS is a graded cellular algebra (Mathas-Hu). Does it have a quasi-hereditary cover (categorifying the full VS)? Relation with Webster’s categorification of VS and Stroppel and Webster’s quiver Schur algebras? This is joint work in progress with Pan and Tubbenhauer. What about k > 3? Yonezawa and I have categorified the slk web spaces using matrix factorizations and also defined a categorical action of U(sln) on these. Hopefully, we can prove the analogue for k > 3 of all the results I have shown you today. Can we get a strictly combinatorial definition of slk–foams (categorification of Cautis-Kamnitzer-Morrison)? 79

The End

THANKS!!! 80