Web calculi in representation theory

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Web calculi in representation theory

Or: the diagrammatic presentation machine Daniel Tubbenhauer

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Joint work with David Rose, Antonio Sartori, Pedro Vaz and Paul Wedrich

August 2015

Daniel Tubbenhauer August 2015 1 / 30

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History of diagrammatic presentations in a nutshell

Rumer, Teller, Weyl (1932), Temperley-Lieb, Jones, Kauffman, Lickorish, Masbaum-Vogel, ... (≥1971):

Uq(sl2)-tensor category generated byC²_q. Kuperberg (1995):

Uq(sl3)-tensor category generated by^V¹_qC³_q∼=C³_q and^V²_qC³_q. Cautis-Kamnitzer-Morrison (2012):

Uq(slN)-tensor category generated by ^V^k_qC^N_q. Sartori (2013),Grant (2014):

Uq(gl_1|1)-tensor category generated by ^V^k_qC^1|1_q . Rose-T. (2015):

Uq(sl2)-tensor category generated bySym^k_qC²_q.Thus,Uq(sl2)-Mod.

Link polynomials:Queffelec-Sartori (2015); “algebraic”:Grant (2015):

Uq(gl_N|M)-tensor category generated by ^V^k_qC^N|M_q . T.-Vaz-Wedrich (2015):

Uq(gl_N|M)-tensor category generated by ^V^k_qC^N|M_q andSym^k_qC^N|M_q . Sartori-T. (maybe! 2015):

Uq(so2N+1,sp_2N,so2N)-tensor categories generated by ^V^kqC^2N(+1)_q .

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1 The story forsl2

Graphical calculus via Temperley-Lieb diagrams The full story forsl2

Proof? Symmetric Howe duality!

2 Exteriorgl_N-web categories Its cousins: theN-webs Proof? Skew Howe duality!

3 As far as we can go in typeAN−1

Even more cousins: the green-redN-webs Proof? Super Howe duality!

4 The machine in action – yet again

What happens in typesBN, CN andDN? This!

Promise: no moreq’s from now on. But you can insert them everywhere if you like.

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The 2-web space

Definition(Rumer-Teller-Weyl 1932)

The 2-web spaceHom2-Web(b,t) is the freeC-vector space generated by non-intersecting arc diagrams withb,t bottom/top boundary points modulo:

Circle

removal: ¹ =−2.

Isotopy relations:

1 1

=

1 1

=

1 1

Daniel Tubbenhauer Graphical calculus via Temperley-Lieb diagrams August 2015 4 / 30

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The 2-web category

Definition(Kuperberg 1995)

The 2-web category 2-Webis the (braided) monoidal,C-linear category with:

Objects are vectors~k = (1, . . . ,1) and morphisms areHom2-Web(~k,~l).

Composition◦:

1 1

◦

1 1

= ¹ ,

1 1

◦

1 1

=

1 1

Tensoring⊗:

1 1

⊗

1 1

=

1 1

1 1 1

1

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Diagrams for intertwiners

Observe that there are (up to scalars) uniqueU(sl2)-intertwiners cap: C²⊗C²։C, cup:C֒→C²⊗C², projectingC²⊗C² ontoCrespectively embeddingCintoC²⊗C².

Letsl2-Modbe the (braided) monoidal,C-linear category whose objects are tensor generated byC². Define a functor Γ : 2-Web→sl2-Mod:

~k = (1, . . . ,1)7→C²⊗ · · · ⊗C²,

1 1

7→cap ,

1 1

7→cup

Theorem(Folklore)

Γ : 2-Web^⊕→sl₂-Modis an equivalence of (braided) monoidal categories.

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The symmetric story

Aredsl2-web is a labeled trivalent graph locally made of

cap_k =

k k

, cup^k =

k k

, m^k+l_k,l =

k+l

k l

, s^k,l_k+l=

k+l

k l

Herek,l,k+l∈ {0,1, . . .}.

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer The full story forsl2 August 2015 7 / 30

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Let us form a category again

Define the (braided) monoidal,C-linear category 2-Webrby using:

Definition

Thered 2-web space Hom2-Webr(~k,~l) is the freeC-vector space generated byred 2-webs modulo the circle removal, isotopies and:

gl_m “ladder”

relations :

l k

l+1 k−1

1 1

−

k l

k+1 l−1

1 1

= (k−l)

l k

Dumbbell

relation : 2

1 1

=−

1 1

+

1 1

2

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Diagrams for intertwiners

Observe that there are (up to scalars) uniqueU(sl2)-intertwiners

cap_k:Sym^kC²⊗Sym^kC²։C, cup^k:C֒→Sym^kC²⊗Sym^kC²,

m^k_k^+l_,l :Sym^kC²⊗Sym^lC²։Sym^k+lC², s^k_k^,l_+l: Sym^k+lC²֒→Sym^kC²⊗Sym^lC² given by projection and inclusion.

Letsl₂-Mods be the (braided) monoidal,C-linear category whose objects are tensor generated bySym^kC². Define a functor Γ : 2-Webr→sl₂-Mods:

~k= (k1, . . . ,km)7→Sym^k¹C²⊗ · · · ⊗Sym^k^mC²,

k k

7→cap_k ,

k k

7→cup^k ,

k+l

k l

7→m^k_k^+l_,l ,

k+l

k l

7→s^k_k^,l_+l

Theorem

Γ : 2-Web^⊕_r →sl₂-Mods is an equivalence of (braided) monoidal categories.

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“Howe” to prove this?

Howe: the commuting actions ofU(glm) andU(glN) on Sym^K(C^m⊗C^N)∼= M

k1+···+km=K

(Sym^k¹C^N⊗ · · · ⊗Sym^k^mC^N)

introduce anU(glm)-actionf on the right term with~k-weight space Sym^~^kC^N. In particular, there is a functorial action

Φ^msym: ˙U(glm)→gl_N-Mods,

~k 7→Sym^~^kC^N, X ∈1~lU(gl_m)1~k 7→f(X)∈Homgl_N-Mods(Sym^~^kC^N,Sym^~^lC^N).

Howe: Φ^m_sym isfull. Or in words:

relations in ˙U(gl_m) + kernel of Φ^m_sym relations ingl_N-Mods.

Daniel Tubbenhauer Proof? Symmetric Howe duality! August 2015 10 / 30

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The diagrammatic presentation machine

Theorem

Define2-Webrsuch there is a commutative diagram U(gl˙ _m) ^Φ

m

sym //

Υ^m

$$

❏❏

❏

gl₂-Mods

2-Webr Γ

99

ss ss ss ss ss

with

Υ^m(Ei1~k)7→

ki ki+1

ki+1−1 ki+1

1 , Υ^m(Fi1~k)7→

ki ki+1

ki−1 ki+1 +1 1

Υ^m gl_m “ladder” relations, ker(Φ^m_sym) dumbbell relation.

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Exempli gratia

Thegl_m “ladder” relations come up as follows:

EF1~k−FE1~k= (k−l)1~k

l k

l+1 k−1

1 1

−

k l

k+1 l−1

1 1

= (k−l)

l k

The dumbbell relation comes up as follows:

C²⊗C²∼=^V²C²⊕Sym²C²∼=C⊕Sym²C²

2

1 1

=−

1 1

+

1 1

2

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It is even better than expected!

The hardestgl_m “ladder” relations, e.g. Serre relations as E_i²Ei+11~k−2EiEi+1Ei1~k+Ei+1E_i²1~k= 0

h k l

h+2 k−1 l−1

1 1

1 a

k

b

−2

h k l

h+2 k−1 l−1

1 1

1

a k

c

+

h k l

h+2 k−1 l−1

1 1

1

a

c

d = 0

do not have to be forced to hold, but are consequences. This pattern repeats in for other web categories.

Morally: web categories have avery economic presentation!

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Replace red by green and add orientations

AgreenN-web is an oriented, labeled, trivalent graph locally made of

m^k+l_k,l =

k+l

k l

, s^k,l_k+l=

k+l

k l

k,l,k+l ∈N

(and some caps, cups and signs that I skip today).

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer Its cousins: theN-webs August 2015 14 / 30

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Let us form a category again

Define the (braided) monoidal,C-linear categoryN-Webg by using:

Definition(Cautis-Kamnitzer-Morrison 2012)

ThegreenN-web space HomN-Webg(~k,~l) is the freeC-vector space generated by greenN-webs modulo isotopies and:

gl_m “ladder”

relations :

l k

l+1 k−1

1 1

−

k l

k+1 l−1

1 1

= (k−l)

l k

Exterior

relation : ^k = 0 , if k>N.

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Diagrams for intertwiners

Observe that there are (up to scalars) uniqueU(glN)-intertwiners

m^k+l_k,l : ^V^kC^N⊗^V^lC^N։^V^k+lC^N , s^k,l_k+l:^V^k^+lC^N֒→^V^kC^N⊗^V^lC^N given by projection and inclusion.

Letgl_N-Mode be the (braided) monoidal,C-linear category whose objects are tensor generated by^V^kC^N. Define a functor Γ :N-Webg→gl_N-Mode:

~k = (k1, . . . ,km)7→^V^k¹C^N⊗ · · · ⊗^V^k^mC^N,

k+l

k l

7→m^k+l_k,l ,

k+l

k l

7→s^k,l_k+l

Theorem(Cautis-Kamnitzer-Morrison 2012)

Γ :N-Web^⊕g →gl_N-Mode is an equivalence of (braided) monoidal categories.

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“Howe” to prove this?

Howe: the commuting actions ofU(glm) andU(glN) on

VK(C^m⊗C^N)∼= M

k₁+···+km=K

(^V^k¹C^N⊗ · · · ⊗^V^k^mC^N)

introduce anU(glm)-actionf on the right term with~k-weight space ^V^~^kC^N. In particular, there is a functorial action

Φ^m_skew: ˙U(gl_m)→gl_N-Mode,

~k7→^V^~^k_qC^N, X ∈1~lU(gl_m)1~k 7→f(X)∈Homgl_N-Mode(^V^~^k_qC^N,^V^~^l_qC^N).

Howe: Φ^m_skewis full. Or in words:

relations in ˙U(glm) + kernel of Φ^m_skew relations ingl_N-Mode.

Daniel Tubbenhauer Proof? Skew Howe duality! August 2015 17 / 30

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Define the diagrams to make this work

Theorem(Cautis-Kamnitzer-Morrison 2012)

DefineN-Webg such there is a commutative diagram U(gl˙ _m) ^Φ

m

skew //

Υ^m

%%

❏❏

gl_N-Mode

N-Webg Γ

88

rr rr rr rr rr r

with

Υ^m(Ei1~k)7→

ki+1 ki

ki+1−1 ki+1

1 , Υ^m(Fi1~k)7→

ki ki+1

ki−1 ki+1 +1 1

Υ^m gl_m “ladder” relations, ker(Φ^m_skew) exterior relation.

Daniel Tubbenhauer Proof? Skew Howe duality! August 2015 18 / 30

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Could there be a pattern?

Agreen-redN-webis a colored, labeled, trivalent graph locally made of

m^k+l_k,l =

k+l

k l

, m^k+l_k,l =

k+l

k l

, m^k+l_k_,1 =

k+ 1

k 1

, m^k+l_k,1 =

k+ 1

k 1

And of course splits and some mirrors as well!

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

Daniel Tubbenhauer Even more cousins: the green-redN-webs August 2015 19 / 30

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The green-red N-web category

Define the (braided) monoidal,C-linear categoryN-Webgrby using:

Definition

Given~k ∈Z^m+n

≥0 ,~l ∈Z^m^′⁺ⁿ^′

≥0 . Thegreen-red N-web space HomN-Webgr(~k,~l) is the freeC-vector space generated byN-webs modulo isotopies and:

gl_m+gl_n

“ladder”

relations

: same as before, but now ingreenandred!

Dumbbell relation : 2

1 1

=

1 1

2 +

1 1

2

Exterior

relation : ^k = 0 , ifk >N.

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Diagrams for intertwiners - Part 4

Observe that there are (up to scalars) uniqueU(glN)-intertwiners

m^k+1_k,1 : ^V^kC^N⊗C^N։^V^k+1C^N, m^k_k⁺¹_,1 :Sym^kC^N⊗C^N ։Sym^k+1C^N plus others as before.

Letgl_N-Modes be the (braided) monoidal,C-linear category whose objects are tensor generated by^V^kC^N,Sym^kC^N. Define a functor Γ :N-Webgr→gl_N-Modes:

~k = (k1, . . . ,km,km+1, . . . ,km+n)7→^V^k¹C^N⊗ · · · ⊗Sym^k^m+nC^N,

k+1

k 1

7→m^k+1_k,1 ,

k+1

k 1

7→m^k+1_k,1 , · · ·

Theorem

Γ :N-Web^⊕_gr→gl_N-Modes is an equivalence of (braided) monoidal categories.

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Super gl

_m|n

Definition

Thegeneral linear superalgebraU(gl_m|n) is generated byHi andFi,Ei subject the some relations, most notably, thesuper relations:

E_m² = 0 =F_m², Hm+Hm+1=FmEm+EmFm, 2EmEm+1Em−1Em=EmEm+1EmEm−1+Em−1EmEm+1Em

+Em+1EmEm−1Em+EmEm−1EmEm+1 (plus an F version).

There is a Howe pair (U(gl_m|n),U(glN)) with~k = (k1, . . . ,km+n)-weight space under theU(gl_m|n)-action on^V^K(C^m|n⊗C^N) given by

Vk1C^N⊗ · · ·^V^k^mC^N⊗Sym^k^m+1C^N⊗ · · · ⊗Sym^k^m+nC^N.

An aside: everything works forgreen-redU(gl_N|M)-webs as well, with the Howe pair (U(glm|n),U(glN|M)).

Daniel Tubbenhauer Proof? Super Howe duality! August 2015 22 / 30

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Define the diagrams to make this work - yet again

Theorem

DefineN-Webgr such there is a commutative diagram U˙q(gl_m|n) ^Φ

m|n

su //

Υ▲^m|n_su▲▲▲▲▲▲▲%%

▲▲

gl_N-Modes

N-Webgr Γ

88

rr rr rr rr rr r

with

Υ^m|n_su (Em1~k)7→

km+1 km

km+1−1 km+1

1 , Υ^m|n_su (Fm1~k)7→

km km+1

km−1 km+1 +1 1

Υ^m|nsu “gl_m|n ladder” relations, ker(Φ^m|nsu ) the exterior relation.

Daniel Tubbenhauer Proof? Super Howe duality! August 2015 23 / 30

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Another meal for our machine

Howe: the commuting actions ofU(so2m) andU(so2N(+1)) on

VK

(C^m⊗C^2N(+1))∼= M

k1+···+kn=K

V^~kC^2N(+1)

introduce anU(so2m)-actionf with~k-weight space^V^~^kC^2N(+1). In particular, there is a functorial action

Φ^m_so: ˙U(so2m)→so_2N(+1)-Mode,

~k 7→^V^~^kC^2N(+1), etc..

Howe: Φ^msois full. Or in words:

relations in ˙U(so2m) + kernel of Φ^mso relations inso_2N(+1)-Mode.

Daniel Tubbenhauer What happens in typesBN,CNandDN? August 2015 24 / 30

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And another one

Howe: the commuting actions ofU(sp_2m) andU(sp_2N) on

VK(C^m⊗C^2N)∼= M

k1+···+kn=K V~kC^2N

introduce anU(sp_2m)-actionf with~k-weight space^V^~^kC^2N. In particular, there is a functorial action

Φ^m_sp: ˙U(sp_2m)→sp_2N-Mode,

~k 7→^V^~^kC^2N, etc.

Howe: Φ^msp isfull. Or in words:

relations in ˙U(sp_2m) + kernel of Φ^msp relations insp_2N-Mode.

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The definition of the diagrams is already determined

Theorem

DefineN-BDWebg such there is a commutative diagram U(so˙ 2m) ^Φ

2m

so //

Υ^m_so

&&

▼▼

▼

so_2N(+1)-Mode

N-BDWebg Γ

77

♥♥

DefineN-CWebgsuch there is a commutative diagram U(sp˙ 2m) ^Φ

2m

sp //

Υ^m_sp

%%

▲▲

sp_2N-Mode

N-CWebg Γ

88

♣♣

♣

Υ^m_so so_2m “ladder” relations, Υ^m_sp sp_2m“ladder” relations etc.

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Green type BCD-webs

Greenwebs in types BN andDN are generated by

k k

,

k+l

k l

,

k+l

k l

,

k k

,

k k

Greenwebs in type CN are generated by

k k

,

k+l

k l

,

k+l

k l

, •

2k

, •

2k

The lanterns reflect the fact that^V^kC^2N is notirreducible in typeCN:

•

2k

slantern:^V^kC^2N։C, •

2k

plantern:C֒→^V^kC^2N

Daniel Tubbenhauer This! August 2015 27 / 30

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Red type BCD-webs

There are also Howe pairs (U(sp_2m),U(so2n(+1))) and (U(so2m),U(sp_2n)) acting now on the symmetric tensors. Guess what comes out:red webs!

Redwebs in type CN are generated by

k k

,

k+l

k l

,

k+l

k l

,

k k

,

k k

Redwebs in types BN andDN are generated by

k k

,

k+l

k l

,

k+l

k l

, •

2k

, •

2k

The lanterns reflect the fact thatSym^kC^2N(+1)isnotirreducible in typesBN,DN:

•

2k

slantern:Sym^2kC^2N ։C, •

2k

plantern:C֒→Sym^2kC^2N

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I do not have tenure. So I have to bore you a bit more.

Some additional remarks.

Homework: feed the machine with yourfavorite duality.

Everything quantizes without too many difficulties. The quantized version sheds new light on HOMFLY-PT, Kauffman and Reshetikhin-Turaev polynomials: their symmetries can beexplainedrepresentation theoretical.

Some parts even work in thenon-semisimple case (e.g. at roots of unities).

The whole approach seems to be amenable to categorification.

Relations to categorifications of the Hecke algebra using Soergel bimodules or categoryO need to be worked out.

This could lead to a categorification of ˙Uq(gl_m|n) (since the “complicated”

super relations are build in the calculus).

A “green-red-foamy” approach could shed additional light on colored Khovanov-Rozansky homologies.

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There is stillmuchto do...

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Thanks for your attention!