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

October 2015

Daniel Tubbenhauer October 2015 1 / 28

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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^k_qC^2N(+1)q .

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1 Some of the first diagrammatic algebras Classical Schur-Weyl duality

Graphical calculus via Temperley-Lieb diagrams The diagrammatic presentation machine

2 The whole story forsl2

Symmetricsl2-webs

Proof? Symmetric Howe duality!

Some cousins

3 Applications

Link invariants `a la Reshetikhin-Turaev Colored Jones and HOMFLY-PT polynomials

Promise: no moreq’s till the very end. But you can insert them everywhere.

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The question we want to solve

The symmetric groupSm inmletters is:

Smis the the group of automorphisms of the set{1, . . . ,m}, Sm=hσ1, . . . , σm−1|Ri,R=







σ²_i = 1, i= 1, . . . ,m−1 σiσjσi =σjσiσj, |i−j|= 1.

σiσj =σjσi, |i−j|>1.

The first description is given “by nature” and explains whyS_m is interesting. The second is a theorem and a “working horse”.

Given a Lie algebrag, we canask the same:

g-Mod category of finite-dimensionalU(g)-modules, g-Mod=h?|??i.

The first description is given “by nature” and explains whyg-Modis interesting.

So, we want the second as well!

Daniel Tubbenhauer Classical Schur-Weyl duality October 2015 4 / 28

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The symmetric group - diagrammatically

The symmetric groupS_m can be described as:

Sm=

*

· · · · · · = 1, = , ^{· · ·} = ^· ^··

+

Similarly forC[Sm].

LetCⁿwith basisv1, . . . ,v_n. ThenC[Sm] acts on (Cⁿ)^⊗m by permuting entries:

· · · · vj1 vji−1 vji vji+1vji+2 vjm

v_j₁ vji−1v_j_i+1 v_j_i v_j_i+2 v_j_m

: (Cⁿ)^⊗m→(Cⁿ)^⊗m.

This is a well-defined action (check relations!).

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The algebra U(gl

_n

)

LetU(gl_n) be the universal enveloping algebra of the Lie algebragl_n.U(gl_n) is given viagenerators and relations:

U(gl_n) =hEi,Fi,Hj|i= 1, . . . ,n−1; j= 1, . . . ,ni/some relations, (the relations are lifts of the relations among the matrices ofgl_n).

Example

Recall thatgl₂is generated by E =

0 1 0 0

, F=

0 0 1 0

, H1=

1 0 0 0

, H2=

0 0 0 1

.

TheC-algebraU(gl₂) consists of words in the symbolsE,F,H1,H2modulo EF−FE =H1−H2

(plus a few other relations).

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C [ S

_m

] is “dual” to U(gl

_n

)

SinceU(gl_n) acts “as matrices” onCⁿ, we can extend it to (Cⁿ)^⊗m via

∆(Ei) = 1⊗Ei+Ei⊗1, ∆(Fi) = 1⊗Fi+Fi⊗1, ∆(Hi) =Hi⊗Hi.

Theorem (Schur 1901)

The actions ofC[Sm] andU(gl_n) on (Cⁿ)^⊗m commute and they generate each other commutant. In particular, they induce an algebra homomorphism

Φ^mSW:C[Sm]։EndU(gl_n)((Cⁿ)^⊗m), Φ^mSW:C[Sm]−^∼⁼→EndU(gl_n)((Cⁿ)^⊗m), ifn≥m, (and of course a “dual version” which we do not need).

In words: Schuralmostgave a diagrammatic generators and relations description of the full subcategorygl₂-Mode ofgl_n-Modtensor generated by the vector representationCⁿ ofU(gl_n).

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The 2-web space

Definition(Rumer-Teller-Weyl 1932)

The 2-web spaceHom2-Web(b,t) is the freeC-vector spacegeneratedby 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 October 2015 8 / 28

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The 2-web category

Definition(Kuperberg 1995)

The2-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 ontoCrespectivelyembeddingCintoC²⊗C².

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

~k = (1, . . . ,1)7→C²⊗ · · · ⊗C²,

1 1

7→cap ,

1 1

7→cup

Theorem(Folklore, Rumer-Teller-Weyl 1932)

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

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The diagrammatic presentation machine

ConsiderC[Sm] as aC-linear category. By Schur-Weyl duality there is afull functor Φ^m_SW:C[Sm]→gl₂-Mode.

Theorem

Define2-Websuch there is a commutative diagram C[Sm] ^Φ

m

SW //

Υ■■^Sm■■■■■■$$

■

gl₂-Mode

2-Web

Γ

99

rr rr rr rr r

with

Υ^S^m

7→

1 1

+

1 1

Υ^S^m circle relation, isotopy relations, ker(Φ^m_SW) isotopy relations

Daniel Tubbenhauer The diagrammatic presentation machine October 2015 11 / 28

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From gl

₂

to sl

₂

Restricting fromgl₂ tosl₂ could increase the number of intertwiners:

U(sl2)⊂U(gl₂) ⇒ HomU(sl2)(M,M^′)⊃HomU(gl₂)(M,M^′).

Note thatC² is self-dual as aU(sl2)-module, butnotas aU(gl₂)-module. We obtain extra diagrams:

1 1

:C²⊗C²→C,

1 1

:C→C²⊗C².

These satisfy the isotopy relations and “fill up the missing” hom-spaces:

Hom_U(gl₂₎(C²⊗C²,C) = 0, but Hom_U(sl₂₎(C²⊗C²,C) =

*

1 1

+

, etc.

Daniel Tubbenhauer The diagrammatic presentation machine October 2015 12 / 28

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The symmetric story

Aredsl2-web is a labeled trivalent graph locallygeneratedby

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 Symmetricsl2 -webs October 2015 13 / 28

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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, e.g.:

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 byprojectionandinclusion.

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, . . . ,k_m)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(gl_m) andU(gl_N) on Sym^K(C^m⊗C^N)∼= M

k1+···+km=K

(Sym^k¹C^N⊗ · · · ⊗Sym^k^mC^N)

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

Φ^msym: ˙U(gl_m)→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! October 2015 16 / 28

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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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No fancy stuff like Karoubi completions needed

Fact:all irreducibleU(sl2)-modules are of the formSym^kC²for somek. Thus, sl2-Mods contains all finite-dimensional representations.

In particular, the Jones-Wenzl projectors of the TL algebra (RTW algebra)

· · ·

JWk = 1

k!

1 k−1

k 1 k−2

1 k−2

.. .

are encoded (and alsoall their relations!).

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As far as we can go in type A

We could also considersl_N instead ofsl2(diagram categoryN-Webr). And ^V^kC^N instead ofSym^kC^N (diagram categoryN-Webg). Or both together (diagram categoryN-Webgr). The graphical calculi for these arevery similar.

Example

5 2 6 1 7

6 6 7 2

5

7 1 8

2 3

5

1 6

greenk!^V^kC^N, redk !Sym^kC^N,

black 1!^V¹C^N ∼=Sym¹C^N∼=C^N.

Daniel Tubbenhauer Some cousins October 2015 20 / 28

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The machine in action again

They are look the same because they are spit out by our machine, e.g.:

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|nsu (Em1~k)7→

km+1 km

km+1−1 km+1

1 , Υ^m|nsu (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 Some cousins October 2015 21 / 28

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Link invariants via representation theory

Color link components withUq(g)-modules. Put the links into a Morse position.

V1

V2

C(q)

C(q)⊗V1⊗V1⊗C(q) V1⊗C(q)⊗V1⊗C(q) V1⊗C(q)⊗C(q)⊗V1

V1⊗V2⊗V2⊗V1

V2⊗V1⊗V2⊗V1

V1⊗V2⊗V2⊗V1

V₁⊗C(q)⊗C(q)⊗V₁ V1⊗C(q)⊗V1⊗C(q) C(q)⊗V1⊗V1⊗C(q)

C(q) εV1

shift shift εV2

RV2⊗V1

RV1⊗V2

ιV2

shift shift ιV1

Theorem (Reshetikhin-Turaev 1990)

The compositeP_~^q

V(1)∈Q(q) is an invariant of(framed, oriented)links.

Daniel Tubbenhauer Link invariants `a la Reshetikhin-Turaev October 2015 22 / 28

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Wait: we have a diagrammatic calculus

Recall that there was an action ofC[Sm] on 2-Web. Thisquantizes:

Υ^S^m

7→

1 1

+

1 1

Υ^H^m

7→ q¹²

|{z}

normalization¹

1

1 1

+q⁻¹

1 1

1

1!

Similarly, our diagrammatic calculusquantizes. The difference is

1 =−2 ¹ =−[2] =−q−q⁻¹.

Theorem (Kauffman 1987)

Using these in the Reshetikhin-Turaev set-up withg=sl2and onlyC²_qas colors gives a combinatorial way to compute the Jones polynomial.

There is a framing shift which I hide, but never mind.

Daniel Tubbenhauer Colored Jones and HOMFLY-PT polynomials October 2015 23 / 28

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Exempli gratia

C²_q

C²_q 1 1

+q⁻¹·

1 1

+q⁻¹·

1 1

+q⁻²·

1 1

P_C^q2

q,C²_q(L) =q((−q−q⁻¹)²+q⁻¹(−q−q⁻¹) +q⁻¹(−q−q⁻¹) +q⁻²(−q−q⁻¹)²)

=q(q+q⁻¹)(q+q⁻³).

This is (up tonormalization) the Jones polynomial of the Hopf link.

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Wait: we have even more diagrammatic calculi

We can quantize the category 2-Webr and obtain a braided monoidal category which enables us to cook up link invariantsdiagrammatically. The braiding is:

k l

=(−1)^kq^−k−^kl²

| {z }

normalization

X

j1,j2≥0 j1−j2=k−l

(−q)^j¹

l k

l+j1−j2 k−j1 +j2

l+j1 k−j1

j1 j2

Theorem

Using these in the Reshetikhin-Turaev set-up withg=sl2andSym^k_qC²_qas colors gives a new, combinatorial way to compute the colored Jones polynomial.

This workscompletely similarfor the categoriesN-Webg,N-Webr andN-Webgr

giving rise to a new way to compute coloredsl_N polynomials for all colors (and thus, colored HOMFLY-PT polynomials).

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Another application: the HOMFLY-PT symmetry

There is also a polynomial calledcolored HOMFLY-PT polynomial

P_λ^a,q(K)∈C(a,q) (K“=”knot). The colorsλare Young diagrams. The whole framework should be seen as the“N→ ∞”-versionof thesl_N Reshetikhin-Turaev approach (a q^N) withλcorresponding to irreducible highest weight module.

From the diagrammatic calculi we obtain:

Corollary (the HOMFLY-PT symmetry)

The colored HOMFLY-PT polynomial satisfies

P_λ^a,q(K) =(−1)^coP_λ^a,qT⁻¹(K), whereco is some constant. Similar for links.

This is arepresentation theoretical explanationof the the HOMFLY-PT symmetry.

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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.

We are working on thetypeB,CandD-versionsand the diagrams work fine (yet, the quantization is complicated).

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

The whole approach seems to beamenableto categorification.

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

This could lead to a categorification ofU˙q(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!