Web bases and categorification

Web bases and categorification

Daniel Tubbenhauer

Joint work with Marco Mackaay and Weiwei Pan

October 2013

1 Categorification

What is categorification?

2 Webs and representations ofUq(sl3) sl3-webs

Connection to representation theory Webs andq-skew Howe duality

3 The Categorification An algebra of foams A graded cellular basis Harvest time

Daniel Tubbenhauer October 2013 2 / 39

What is categorification?

Forced to reduce this presentation to one sentence, the author would choose:

Interesting integers are shadows of richer structures in categories.

The basic idea can be seen as follows. Take a“set-based”structureS and try to find a“category-based”structureC such thatS is just a shadow ofC.

Categorification, which can be seen as“remembering” or “inventing”information, comes with an “inverse” process calleddecategorification, which is more like

“forgetting” or “identifying”.

Note that decategorification should beeasy.

Exempli gratia

Examples of the pair categorification/decategorification are:

Bettinumbers of a manifoldM ^categorify //

decat=rank(·)

oo Homology groups

Polynomials inZ[q,q⁻¹] ^categorify //

decat=χgr(·)

oo complexes of gr.VS

The integersZ

categorify

//

decat=K0(·)

oo K−vector spaces

AnA−module ^categorify //

decat=K₀^⊕(·)⊗_ZA

oo additive category

Usually thecategorified worldis much moreinteresting.

Today categorification = Grothendieck group!

Daniel Tubbenhauer What is categorification? October 2013 4 / 39

Kuperberg’s sl

-webs

Definition(Kuperberg)

Asl3-webw is anoriented trivalent graph, such that all vertices are either sinks or sources. The boundary∂w ofw is asign stringS = (s1, . . . ,sn)under the conventionsi= + iff the orientation is pointing in andsi =−iff the orientation is pointing out.

Example

+ − + − + + +

Kuperberg’s sl

-webs

Definition(Kuperberg)

TheC(q)-web spaceWS for a given sign stringS = (±, . . . ,±) is generated by {w |∂w =S}, wherew is a web, subject to the relations

= [3]

= [2]

= +

Here [a] = ^q_q−q^a−q−1^−a =q^a−1+q^a−3+· · ·+q^−(a−1) is thequantum integer.

Daniel Tubbenhauer sl3-webs October 2013 6 / 39

Kuperberg’s sl

-webs

Example

0 0 0 0 0 +1 −1

Webs can becolouredwith flow lines. At the boundary, the flow lines can be represented by astate stringJ. By convention, at thei-th boundary edge, we set j_i =±1 if the flow line is oriented downward/upward andj_i = 0, if there is no flow line. SoJ = (0,0,0,0,0,+1,−1) in the example.

Given a web with a flowwf, attribute aweightto each trivalent vertex and each

The quantum algebra U

(sl

)

Definition

Forn∈N_>1thequantum special linear algebraU_q(sln) is the associative, unital C(q)-algebra generated byK_i^±1andE_i andF_i, fori= 1, . . . ,n−1, subject to the relations

KiKj =KjKi, KiK_i⁻¹=K_i⁻¹Ki = 1, EiFj−FjEi=δi,j

Ki−K_i⁻¹ q−q⁻¹ , KiEj =q^(ǫi,αj)EjKi, K_iF_j =q^−(ǫi,αj)F_jK_i,

E_i²Ej−(q+q⁻¹)EiEjEi+EjE_i²= 0, if |i−j|= 1, F_i²F_j−(q+q⁻¹)FiF_jF_i+F_jF_i²= 0, if |i−j|= 1.

Daniel Tubbenhauer Connection to representation theory October 2013 8 / 39

Representation theory of U

(sl

)

A sign stringS = (s1, . . . ,sn) corresponds to tensors VS =Vs1⊗ · · · ⊗Vsn,

whereV+ is the fundamentalUq(sl3)-representation andV− is its dual, and webs correspond tointertwiners.

Theorem(Kuperberg)

W_S ∼=HomUq(sl₃)(C(q),V_S)∼=InvUq(sl₃)(VS)

In fact, the so-called spider category of all webs modulo the Kuperberg relations is equivalentto the representation category ofU_q(sl3).

As a matter of fact, thesl3-webs without internal circles, digons and squares form abasisB_S, calledweb-basis, ofW_S!

Representation theory of U

(sl

)

Theorem(Khovanov, Kuperberg)

Pairs of signS and a state stringsJ correspond to the coefficients of the web basis relative totensors of the standard basis{e₋₁^± ,e₀^±,e₊₁^±} ofV±.

Example

wt=−2

0 0 0 0 0 +1 −1

+ − + − + + +

wt=−4

0 0 0 0 0 +1 −1

+ − + − + + +

w_S =· · ·+ (q⁻²+q⁻⁴)(e₀⁺⊗e₀⁻⊗e₀⁺⊗e₀⁻⊗e₀⁺⊗e₋₁⁺ ⊗e₊₁⁺)± · · ·.

Daniel Tubbenhauer Connection to representation theory October 2013 10 / 39

Skew Howe-duality

The natural actions ofGLk andGLnon

^p

(C^k ⊗Cⁿ) areHowe dual(skew Howe duality).

Thisimpliesthat

InvSL_k(Λ^p1(C^k)⊗ · · · ⊗Λ^pn(C^k))∼=W(p1, . . . ,pn),

whereW(p1, . . . ,p_n) denotes the (p1, . . . ,p_n)-weight space of the irreducible GLn-moduleW_(k^ℓ₎, ifn=kℓ.

The idempotent version

For eachλ∈Zⁿ⁻¹ adjoin anidempotent1λ (think: projection to theλ-weight space!) toUq(sln) and add the relations

1λ1µ=δλ,ν1λ, Ei1λ= 1λ+αiEi, F_i1λ= 1λ−αiF_i,

K_±i1λ=q^±λi1λ (noK^′s anymore!).

Definition

Theidempotented quantum special linear algebrais defined by U(sl˙ n) = M

λ,µ∈Zⁿ

1λU_q(sln)1µ.

Daniel Tubbenhauer Webs andq-skew Howe duality October 2013 12 / 39

En(c)hanced sign strings

Definition

Anenhanced sign sequenceis a sequenceS = (s1, . . . ,sn) withsi∈ {◦,−,+,×}, for alli= 1, . . . ,n. The correspondingweightµ=µS ∈Λ(n,d) is given by the rules

µi =











0, ifs_i =◦, 1, ifsi = +, 2, ifs_i =−, 3, ifsi =×.

Let Λ(n,d)3⊂Λ(n,d) be the subset of weights with entries between 0 and 3.

Note that 1 corresponds to the ˙U(sl3)-representationV+, 2 to its dualV− and 0,3 to the trivial ˙U(sl3)-representation.

The sl

-webs form a ˙ U(sl

)-module

Wedefinedan actionφof ˙U(sln) onW₍₃^ℓ₎=L

S∈Λ(n,n)3WS by

1λ7→

λ1 λ2 λn

E_i1λ, F_i1λ7→

λ1 λi−1 λi λi+1

λi±1 λi+1∓1

λi+2 λn

We use the convention that vertical edges labeled 1 are oriented upwards, vertical edges labeled 2 are oriented downwards and edges labeled 0 or 3 are erased. The hard part was to show that this iswell-defined.

Daniel Tubbenhauer Webs andq-skew Howe duality October 2013 14 / 39

Exempli gratia

E11(22)7→

2 2

3 1

F2E11(121)7→

1 2 1

2 0 2

Very nice bases of W

The ˙U(sln)-moduleW₍₃^ℓ₎ has different bases. But there are two particularnice ones, called Lusztig-Kashiwara’slower and upper global crystal basisBT ={bT} andB^T ={b^T}(sometimes also calledcanonical and dual canonicalbasis), indexed by standard tableauxT ∈Std((3^ℓ)). One of its nice properties is for example

b_T =x_T+X

τ≺T

δτT(q)

| {z }

∈Z[q]

x_τ

Thisgives, underq-skew Howe duality, a upper and lower global crystal basis of the invariantU_q(sl3)-tensors.

In contrast to the tensor basisxT, which iseasyto write down, butlacksa good behavior, the lower and upper global crystal bases arehardto write down, but havea good behavior.

Daniel Tubbenhauer Webs andq-skew Howe duality October 2013 16 / 39

An intermediate crystal basis

Leclerc and Toffin gave anintermediatecrystal basis of ˙U(sln)-modules, denoted byLT_T ={A_T}, by the rule

A_T =F_i^(r_s^s)· · ·F_i^(r₁¹⁾vΛ, withF^(k) = F^k [k]!,

where the string ofF^′s is obtained by an explicit, combinatorial algorithm from the tableauT. They showed that the crystal basesbT and the tensor basisxT are related by aunitriangularmatrix

AT =xT+X

τ≺T

ατT(q)xτ andbT =AT+X

S≺T

βST(q)AS,

with certain coefficientsατT(q)∈N[q,q⁻¹] andβST(q)∈Z[q,q⁻¹].

An intermediate crystal basis

Proposition(Mackaay, T)

The Kuperberg web basisBS is Leclerc-Toffin’s intermediate crystal basis under q-skew Howe duality, i.e.

LT_T ={F_i^(r_s^s)· · ·F_i^(r₁¹⁾v₃^ℓ |T ∈Std((3^ℓ))}7−→^sHDLT_S. (NoK’s andE’s anymore!)

Thus, theBS is agoodcandidate for categorification (can be written down explicitlyandhas (some) good properties!).

Daniel Tubbenhauer Webs andq-skew Howe duality October 2013 18 / 39

Exempli gratia

Example

w = ! T =

1 2 4 2 3 5 4 6 6

FormT we obtain the string

LT(w) =F1F2F₃⁽²⁾F2F1F4F3F2F₅⁽²⁾F₄⁽²⁾F₃⁽²⁾.

Exempli gratia

LT(w)v(3³) =F1F2F₃⁽²⁾F2F1F4F3F2F₅⁽²⁾F₄⁽²⁾F₃⁽²⁾v₍₃³₎

F1

1 2 1 2 1 2

F2

2 1 1 2 1 2

F₃⁽²⁾

2 2 0 2 1 2

F2

2 2 2 0 1 2

F1

2 3 1 0 1 2

F4

3 2 1 0 1 2

F₃⁽²⁾

3 3 1 2 0 0

F₄⁽²⁾

3 3 1 0 2 0

F₅⁽²⁾

3 3 1 0 0 2

F2

3 2 2 0 0 2

F3

3 2 1 1 0 2

3 3 3 0 0 0

Daniel Tubbenhauer Webs andq-skew Howe duality October 2013 20 / 39

Please, fasten your seat belts!

Let’scategorifyeverything!

sl

-foams

Apre-foamis a cobordism with singular arcs between two webs. Composition consists of placing one pre-foam ontopof the other. The following are called the zipand theunziprespectively.

They havedotsthat can movefreelyabout the facet on which they belong, but we donotallow dot to cross singular arcs.

Afoamis a formalC-linear combination of isotopy classes of pre-foams modulo the following relations.

Daniel Tubbenhauer An algebra of foams October 2013 22 / 39

The foam relations ℓ = (3 D , NC , S , Θ)

= 0 (3D)

=− − − (NC)

= = 0, =−1 (S)

α β

δ =







1, (α, β, δ) = (1,2,0) or a cyclic permutation,

−1, (α, β, δ) = (2,1,0) or a cyclic permutation, 0, else.

(Θ)

Involution on webs and closed webs

Definition

There is aninvolution^∗ on the webs.

Aclosed webis defined by closing of two webs.

Aclosed foamis a foam from∅ to a closed web.

Daniel Tubbenhauer An algebra of foams October 2013 24 / 39

The sl

-foam category

Foam3is thecategory of foams, i.e.objectsare websw andmorphismsare foams F between webs. The category isgradedby theq−degree

qdeg(F) =χ(∂F)−2χ(F) + 2d+b,

whered is the number of dots andb is the number of vertical boundary components. Thefoam homologyof a closed webw is defined by

F(w) =Foam3(∅,w).

F(w) is a graded, complex vector space, whoseq-dimension can be computed by theKuperberg bracket(that is counting all flows onw and their weights).

The sl

-web algebra

Definition(MPT)

LetS = (s1, . . . ,sn). Thesl3-web algebraKS is defined by KS = M

u,v∈BS

uKv,

with

uKv =F(u^∗v){n}, i.e. all foams : ∅ →u^∗v.

Multiplication is defined as follows.

uKv1⊗v2Kw →uKw

is zero, ifv16=v2. Ifv1=v2, use themultiplication foamm_v, e.g.

Daniel Tubbenhauer An algebra of foams October 2013 26 / 39

The sl

-web algebra

Theorem(s)(MPT)

The multiplication iswell-defined, associative and unital. The multiplication foam m_v hasq-degreen. Hence,K_S is a finite dimensional, unital and graded algebra.

Moreover, it is agraded Frobenius algebra.

Higher representation theory

Moreover, forn=d= 3^k we define

W₍₃^k₎ = M

µs∈Λ(n,n)3

WS

on thelevelof webs and on the levelof foams we define W₍₃^(p)k)= M

µs∈Λ(n,n)3

K_S−(p)Mod_gr.

With this constructions we obtain our firstcategorificationresult.

Theorem(MPT)

K0(W₍₃^k₎)⊗_Z[q,q⁻¹]Q(q)∼=W₍₃^k₎ andK₀^⊕(W₍₃^pk))⊗_Z[q,q⁻¹]Q(q)∼=W₍₃^k₎.

Daniel Tubbenhauer An algebra of foams October 2013 28 / 39

Cellular algebras have a “simple” representation theory

Definition(Graham-Lehrer, Hu-Mathas)

Agraded cellular basisc_st^λ of a graded algebraAis a basis withnicestructure coefficients (and otherniceproperties that we do not need today), i.e.

ac_st^λ = X

u∈T(λ)

ra(s,u)c_ut^λ (modA^⊲λ),

where theλ’s are from a poset (P,⊲) andT(λ) is finite for allλ∈P.

Theorem(Graham-Lehrer, Hu-Mathas)

Forλ∈P one canexplicitly(using the structure coefficients) define the graded cell moduleC^λ. SetD^λ=C^λ/rad(C^λ) and P0={λ∈P|D^λ6= 0}. Then the set{D^λ{k} |λ∈P0, k ∈Z}is acompleteset of pairwisenon-isomorphic

The approach

Recall that the intermediate crystal basis satisfies AT =xT+X

τ≺T

ατT(q)xτ andbT=AT+X

S≺T

βST(q)AS.

Idea: Ifq-skew Howe duality can be used to obtain from the ˙U(sln)-moduleW₍₃^ℓ₎ the intermediate crystal basis on the level of webs, then categorifiedq-skew Howe duality can be used to obtain a cellular basis from acategorifiedintermediate crystal basis on thelevel of foams!

Daniel Tubbenhauer A graded cellular basis October 2013 30 / 39

Connection to U

(sl

)

Khovanov and Lauda’s diagrammatic categorification of ˙Uq(sln), denoted U(sln), is alsorelatedto our framework! Roughly, it consist of string diagrams of the form

i j

λ:EiEj1λ⇒ EjEi1λ{(αi, αj)},

i

λ−αi λ:Fi1λ⇒ Fi1λ{αⁱⁱ}

with a weightλ∈Zⁿ⁻¹and suitable shifts and relations like

i j

λ =

i j

λ and

i j

λ =

i j

λ, ifi6=j.

sl

-foamation

We define a 2-functor

Ψ :U(sln)→ W₍₃^(p)k)

calledfoamation, in the following way.

On objects:The functor is defined by sending anslk-weightλ= (λ1, . . . , λk−1) to an object Ψ(λ) ofW₍₃^(p)k) by

Ψ(λ) =S, S = (a1, . . . ,ak),ai ∈ {0,1,2,3}, λi =ai+1−ai, Xk

i=1

ai= 3^k.

On morphisms:The functor on morphisms is by glueing the ladder webs from before on top of thesl3-webs inW₍₃^k₎.

Daniel Tubbenhauer A graded cellular basis October 2013 32 / 39

sl

-foamation

On 2-cells:We define

i,λ

7→

λi

λi+1

i,λ

7→

λi

λi+1

i,i,λ

7→ −

λi

λi+1

i,i+1,λ

7→ (−1)^λi+1

λi

λi+1

λi+2

i+1,i,λ

7→

λi

λi+1

λi+2

The idea!

Letλ∈Λ(n,n)⁺ be a dominant weight. Define thecyclotomic KL-R algebraRλ

to be the subquotient ofU(sln) defined by the subalgebra ofonly downward (only F’s!)pointing arrows modulo the so-calledcyclotomic relationsand set

Vλ=Rλ−(p)Mod_gr. The cyclotomic KL-R algebraRλ is isomorphic to a certain cyclotomic Hecke algebraHλ of typeA.

Theorem(s)(MPT)

There exists an equivalence of categoricalU(sln)-representations Φ :V₍₃^(p)k) → W₍₃^(p)k).

Idea(T)

Thecombinatoricscan beeasierworked out in the cyclotomic Hecke algebraH_λ, while thetopologyis easierin our framework. Use foamation to pull Hu-Mathas graded cellular basis fromH_λ toK_S.

Daniel Tubbenhauer A graded cellular basis October 2013 34 / 39

A growth algorithm for foams

Definition(T)

Given a pair of a sign string and a state string (S,J), the corresponding

3-multipartition~λand two Kuperberg websu,v ∈BS that extendJ tofu andfv

receptively. We define afoamby F_T(u^~_~^λ

fu),~T(v_fv)= Fσu

|{z}

Topology

e(~λ)

|{z}

Idempotent

d(~λ)

| {z }

Dots

F_σ^∗_v

|{z}

Topology

.

Theorem(T)

The growth algorithm for foams iswell-defined, theonlyinput data are webs and flows on webs, worksinductivelyand gives agraded cellular basis ofK_S.

Exempli gratia

Every web has a graded cellular basisparametrisedby flow lines.

wt=3

wt=1

wt=1

wt=−1

wt=−1

wt=−3

q−deg=0

q−deg=2

q−deg=2

q−deg=4

q−deg=4

q−deg=6

That these foams arereallya graded cellular basis follows from our theorem. Note that the Kuperberg bracket gives [2][3] =q⁻³+ 2q⁻¹+ 2q+q³.

Daniel Tubbenhauer A graded cellular basis October 2013 36 / 39

Categorification and crystal bases

Recall our first categorification result, i.e.

K0(W₍₃^k₎)⊗_Z[q,q⁻¹_]C(q)∼=W₍₃^k₎ andK₀^⊕(W₍₃^pk))⊗_Z[q,q⁻¹_]C(q)∼=W₍₃^k₎. A natural question is how do the two nice bases, i.e. the lower{bT} and the upper{b^T}global crystal basis, ofW(3^k) show upinK0(W(3^k))⊗_Z[q,q⁻¹]Q(q) or K₀^⊕(W₍₃^pk))⊗_Z[q,q⁻¹]Q(q)?

Recall that a graded cellular basis{c_st^λ} gives riseto a set of graded cell modules {C^λ}, their simple heads{D^λ=C^λ/rad(C^λ)}and the corresponding projective covers{C_p^λ} and{D_p^λ}.

Categorification and crystal bases

Theorem(T)

We haveψ([D^T]) =bT andψp([D_p^T]) =b^T under the two isometries

ψ:K0(W₍₃^k₎)⊗_Z[q,q⁻¹_]Q(q)→W₍₃^k₎ andψp:K₀^⊕(W₍₃^pk))⊗_Z[q,q⁻¹_]Q(q)→W₍₃^k₎, that is the simple headsD^T of the cell modules C^T (who give a complete list of all simpleKS-modules)categorifythe lower global crystal basisbT and their projective coversD_p^T (who give a complete list of all projective, irreducible KS-modules)categorifythe upper global crystal basisb^T.

Daniel Tubbenhauer Harvest time October 2013 38 / 39

There is stillmuchto do...

Thanks for your attention!

Daniel Tubbenhauer Harvest time October 2013 39 / 39