sl 3 -web bases, intermediate crystal bases and categorification

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1 sl 3 -web bases, intermediate crystal bases and categorification Daniel Tubbenhauer Joint work with Marco Mackaay and Weiwei Pan October 2013 F F (2) F F (2) 3 F (2) Daniel Tubbenhauer October / 43

2 1 Categorification What is categorification? 2 Webs and representations of U q (sl 3 ) sl 3 -webs Intermediate crystals q-skew Howe duality and growing of webs Growing of flows on webs 3 The Categorification An algebra of foams Growing of foams Foamation and the HM-basis Daniel Tubbenhauer October / 43

3 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 structure S and try to find a category-based structure C such that S is just a shadow of C. Categorification, which can be seen as remembering or inventing information, comes with an inverse process called decategorification, which is more like forgetting or identifying. Note that decategorification should be easy. Daniel Tubbenhauer What is categorification? October / 43

4 Exempli gratia Examples of the pair categorification/decategorification are: Bettinumbers of a manifold M categorify decat=rank( ) Homology groups Polynomials in Z[q,q 1 ] categorify decat=χ gr( ) complexes of gr.vs The integers Z categorify decat=k 0( ) K vector spaces An A module categorify decat=k 0 ( ) ZA additive category Usually the categorified world is much more interesting. Today decategorification = Grothendieck group! Daniel Tubbenhauer What is categorification? October / 43

5 Kuperberg s sl 3 -webs Definition(Kuperberg) A sl 3 -web w is an oriented trivalent graph, such that all vertices are either sinks or sources. The boundary w of w is a sign string S = (s 1,...,s n ) under the convention s i = + iff the orientation is pointing in and s i = iff the orientation is pointing out (we also need 0,3 later - but they are not drawn). Example Daniel Tubbenhauer sl 3 -webs October / 43

6 Kuperberg s sl 3 -webs Definition(Kuperberg) The C(q)-web space W S for a given sign string S = (±,...,±) is generated by {w w = S}, where w is a web, subject to the relations = [3] = [2] = + Here [a] = qa q a q q 1 = q a 1 +q a 3 + +q (a 1) is the quantum integer. Daniel Tubbenhauer sl 3 -webs October / 43

7 Representation theory of U q (sl 3 ) A sign string S = (s 1,...,s n ) corresponds to tensors V S = V s1 V sn, where V + is the fundamental U q (sl 3 )-representation and V is its dual, and webs correspond to intertwiners. Theorem(Kuperberg) W S = HomUq (sl 3)(C(q),V S ) = Inv Uq (sl 3)(V S ) In fact, the so-called spider category of all webs modulo the Kuperberg relations is equivalent to the representation category of U q (sl 3 ). As a matter of fact, the sl 3 -webs without internal circles, digons and squares form a basis B S, called web basis, of W S! Daniel Tubbenhauer sl 3 -webs October / 43

8 Kuperberg s sl 3 -webs Example Webs can be coloured with flow lines. At the boundary, the flow lines can be represented by a state string J. By convention, at the i-th boundary edge, we set j i = ±1 if the flow line is oriented downward/upward and j i = 0, if there is no flow line. So J = (0,0,0,0,0,+1, 1) in the example. Given a web with a flow w f, attribute a weight to each trivalent vertex and each arc in w f and take the sum. The weight of the example is 4. Daniel Tubbenhauer sl 3 -webs October / 43

9 Representation theory of U q (sl 3 ) Theorem(Khovanov-Kuperberg) Pairs of sign S and a state strings J correspond to the coefficients of the web basis relative to tensors of the standard basis {e± 1,e0 ±,e+1 ± } of V ±. Example wt= wt= 4 w S = +(q 2 +q 4 )(e 0 + e 0 e 0 + e 0 e 0 + e +1 + e 1 + )±. Daniel Tubbenhauer sl 3 -webs October / 43

10 What kind of basis is B S? Theorem(Khovanov-Kuperberg) Given (S,J), we have (with v = q 1 and es J = ej1 s 1 es jn n ) w J S = ej S + J <J c(s,j,j )e J S for c(s,j,j ) N[v,v 1 ]. In general we have B S dcan(w S ), but the web basis is bar-invariant. Theorem(MPT) We proved, by categorification, that the change-of-basis matrix from Kuperberg s web basis B S to the dual canonical basis dcan(w S ) is unitriangular. Question: The web basis B S is a somehow special basis of W S. But it is not the dual canonical. So what kind of basis is it? Daniel Tubbenhauer sl 3 -webs October / 43

11 The quantum algebra U q (sl d ) Definition For d N >1 the quantum special linear algebra U q (sl d ) is the associative, unital C(q)-algebra generated by K ±1 i and E i and F i, for i = 1,...,d 1, subject to some relations (that we do not need today). Definition(Beilinson-Lusztig-MacPherson) For each λ Z d 1 adjoin an idempotent 1 λ (think: projection to the λ-weight space!) to U q (sl d ) and add some relations, e.g. 1 λ 1 µ = δ λ,ν 1 λ and K ±i 1 λ = q ±λi 1 λ (no K s anymore!). The idempotented quantum special linear algebra is defined by U(sl d ) = λ,µ Z d 1 1 λ U q (sl d )1 µ. Daniel Tubbenhauer Intermediate crystals October / 43

12 Intermediate crystals Let d = 3l and let V Λ be the irreducible U(sl d )-module of highest weight Λ = (3 l ). Kashiwara-Lusztig s lower global crystal (or canonical) basis can(v Λ ) = {b T T Std(3 l )} has nice properties, but is in very hard to find. Leclerc and Toffin have defined an intermediate crystal basis B Λ of V Λ by an explicit algorithm that can be used to compute can(v Λ ) inductively, i.e. B Λ has some nice properties, but is still trackable enough to be written down. Example (with l = 3) T = LT(T) = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3 v Λ. Daniel Tubbenhauer Intermediate crystals October / 43

13 Sitting in-between {b T } and {x T } The intermediate crystal basis sits in-between the canonical can(v Λ ) and the tensor basis {x T Λ l q (Cd q ) 3 V Λ T Col(3 l )}. Theorem(Leclerc-Toffin) We have (for T Col(3 l ), T Std(3 l )) LT(T) = x T + T T α T (v)x T and b T = LT(T)+ T T β T T(v)LT(T ) with certain α T (v) N[v,v 1 ] and β T T(v) Z[v,v 1 ] (with v = q 1 ). Moreover, the intermediate crystal basis is bar-invariant. We have seen this before! Daniel Tubbenhauer Intermediate crystals October / 43

14 An instance of q-skew Howe duality The commuting actions of U(sl d ) and U(sl 3 ) on Λ q(c d q) 3 = Λ q (C d C 3 ) = Λ q(c 3 q) d introduce an U(sl d )-action on Λ q(c 3 q) d and an U(sl 3 )-action on Λ q(c d q) 3. Here n Λ q(c l q) = Λ k q(c l q) k=1 and all the Λ k q(c l q) are irreducible U(sl l )-representations. Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

15 The sl 3 -webs form a U(sl d )-module We defined an action φ of U(sl d ) on W (3 l ) = S Λ(n,n) 3 W S by 1 λ λ 1 λ 2 λ n λ i±1 λ i+1 1 E i 1 λ, F i 1 λ λ 1 λ i 1 λ i λ i+1 λ i+2 λ n 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. Think: 0,3 indicates the trivial, 1 the V + and 2 the V -representation. Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

16 Exempli gratia 3 1 E 1 1 (22) F 2 E 1 1 (121) Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

17 An intermediate crystal basis Proposition(MT) The Kuperberg web basis B S is Leclerc-Toffin s intermediate crystal basis under q-skew Howe duality, i.e. LT(T) = {F (rs) i s (No K s and E s anymore!) F (r1) i 1 v 3 l T Std(3 l )} shd ws. J Corollary The change-of-basis matrix from Kuperberg s web basis B S to the dual canonical basis dcan(w S ) is unitriangular and w J S = ej S + J <J c(s,j,j )e J S for c(s,j,j ) N[v,v 1 ]. Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

18 Exempli gratia Example w = T = From T we obtain the string LT(T) = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3. Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

19 Exempli gratia LT(T)v 3 3 = F 1 F 2 F (2) 3 F 2 F 1 F 4 F 3 F 2 F (2) 5 F (2) 4 F (2) 3 v F F F (2) F F F F F F (2) F (2) F (2) Daniel Tubbenhauer q-skew Howe duality and growing of webs October / 43

20 3-multipartitions The growth algorithm for webs can be read as webs are standard tableaux. So what are flows on webs and what is the weight of these flows? Observation Column-strict tableaux T Col(3 l ) are 3-multipartitions ,, Question: The flows and their weights encode the coefficients of the web basis B S in terms of the tensor basis {x T T Col(3 l )}. Shouldn t it be possible to encode both using fillings of 3-multipartitions? Daniel Tubbenhauer Growing of flows on webs October / 43

21 3-multitableaux and degrees Definition Let T Std( λ) be a (filled with numbers from {1,...,k}) 3-multitableau T = (T 1,T 2,T 3 ). For j {1,...,k} let N j denote the set of all nodes that are filled with the number j and let T j denote the 3-multitableau obtained from T by removing all nodes with entries > j and set 0, if N j = 1, deg( T j ) = A k N ( T j ) R k N ( T j ) a with a = 1, if N j = 2, 3, if N j = 3, Define deg( T) by deg BKW ( T) k = deg( T j ). j=1 For the experts: This is also known as Brundan-Kleshchev-Wang s degree. Daniel Tubbenhauer Growing of flows on webs October / 43

22 Flows and 3-multitableaux Theorem(T) A suitable T Std( λ) gives rise to a flow on a web via an extended growth algorithm. Moreover, one can obtain all flows on all webs in this way and deg BKW ( T) = k j=1 deg( T j ) is exactly minus the weight of the flow. Although the algorithm is completely explicit, I do not define it today, since there are several rules that one has to follows. I give an example instead. Only keep in mind: Webs are standard tableaux and flows on webs are 3-multitableaux. Daniel Tubbenhauer Growing of flows on webs October / 43

23 Exempli gratia , , Daniel Tubbenhauer Growing of flows on webs October / 43

24 Please, fasten your seat belts! Let s categorify everything! Daniel Tubbenhauer Growing of flows on webs October / 43

25 sl 3 -foams A pre-foam is a cobordism with singular arcs between two webs. Composition consists of placing one pre-foam on top of the other. The following are called the zip and the unzip respectively. They have dots that can move freely about the facet on which they belong, but we do not allow dot to cross singular arcs. A foam is a formal C-linear combination of isotopy classes of pre-foams modulo the following relations. Daniel Tubbenhauer An algebra of foams October / 43

26 The foam relations l = (3D,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) Adding a closure relation to l suffice to evaluate foams without boundary! Daniel Tubbenhauer An algebra of foams October / 43

27 Involution on webs and foams Definition There is an involution on the webs and foams. That is for webs and for foams = A closed foam is a foam from to a closed web u v. Daniel Tubbenhauer An algebra of foams October / 43

28 The sl 3 -foam category Foam 3 is the category of foams, i.e. objects are webs w and morphisms are foams F between webs. The category is graded by the q degree deg q (F) = χ( F) 2χ(F)+2d +b, where d is the number of dots and b is the number of vertical boundary components. The foam homology of a closed web w is defined by F(w) = Foam 3 (,w). F(w) is a graded, complex vector space, whose q-dimension can be computed by the Kuperberg bracket (that is counting all flows on w and their weights). Daniel Tubbenhauer An algebra of foams October / 43

29 The sl 3 -web algebra Definition(MPT) Let S = (s 1,...,s n ). The sl 3 -web algebra K S is defined by K S = uk v, u,v B S with uk v = F(u v){n}, i.e. all foams: u v. Multiplication is defined as follows. uk v1 v2 K w u K w is zero, if v 1 v 2. If v 1 = v 2, use the multiplication foam m v, e.g. Daniel Tubbenhauer An algebra of foams October / 43

30 The sl 3 -web algebra Theorem(s)(MPT) The multiplication is well-defined, associative and unital. The multiplication foam m v has q-degree n. Hence, K S is a finite dimensional, unital and graded algebra. Moreover, it is a graded Frobenius algebra. Daniel Tubbenhauer An algebra of foams October / 43

31 Higher representation theory Moreover, for n = d = 3l we define W (3 l ) = µ s Λ(n,n) 3 W S on the level of webs and on the level of foams we define W (p) (3 l ) = K S (p)mod gr. µ s Λ(n,n) 3 With this constructions we obtain our first categorification result. Theorem(MPT) K 0 (W (3l )) Z[q,q 1 ] Q(q) = W (3l ) and K 0 (Wp (3 l ) ) Z[q,q 1 ] Q(q) = W (3l ). Daniel Tubbenhauer An algebra of foams October / 43

32 Categorification of the LT-algorithm As a remainder, the LT-algorithm gives LT(T) = x T + T T α T (v)x T. Thus, we need column-strict tableaux and 3-multitableaux (for x T and α T (v)). What do we expect to gain? Since Leclerc-Toffin also showed b T = LT(T)+ β T T(v)LT(T ) T T we expect that we get a method to compute the projective indecomposable of K S, since they should decategorify to the dual canonical basis. Daniel Tubbenhauer Growing of foams October / 43

33 This case is a non-trivial idempotent - the web is not a dual canonical web. Daniel Tubbenhauer Growing of foams October / 43 A picture is worth thousand words 3-multitableau: topology of foam categorifies α T (v) column-strict tableau: certain idempotent categorifies x T 3-multitableau: topology of foam categorifies α T (v)

34 A growth algorithm for foams Definition(T) Given a pair of a sign string and a state string (S,J), the corresponding 3-multipartition λ and two Kuperberg webs u,v B S that extend J to f u and f v receptively. We define a foam by Theorem(T) F λ T(ufu ), T(v = F fv ) σ }{{} u e( λ) }{{} Topology Idempotent d( λ) }{{} Dots F σ v }{{} Topology The growth algorithm for foams is well-defined, the only input data are webs and flows on webs, works inductively and gives a graded cellular basis of K S.. Daniel Tubbenhauer Growing of foams October / 43

35 Exempli gratia We have two webs with flows for the pair S = (+,,+, ) and J = (0,0,0,0), i.e. two either nested or non-nested circles without flow In this case the tableaux and the 3-multipartitions are the same, that is T = ( ) λ =,, How to assign a dotted idempotent e( λ)d( λ) that is suitable for both webs? Daniel Tubbenhauer Growing of foams October / 43

36 I do not have time but very roughly: In order to define the topology F σu,f σ v one let S n act on T λ by permuting entries until it looks like the one from the extended growth algorithm. Then use a certain zipping -foam for each transposition. Daniel Tubbenhauer Growing of foams October / 43 Exempli gratia ( 1 2 T λ = 3, 4, ) LT( λ) = F 1 F 3 F 2 F 2 F 1 F 3 F 2 and define a dotted idempotent e( λ)d( λ) by applying q-skew Howe duality and spread dots based on addable nodes. LT( λ)v 3 2 e( λ)d( λ) = LT( λ)v 3 2

37 Connection to U q (sl d ) Khovanov and Lauda s diagrammatic categorification of U q (sl d ), denoted U(sl d ), is also related to our framework! Roughly, it consist of string diagrams of the form i λ: E i E j 1 λ E j E i 1 λ {(α i,α j )}, j λ α i λ: F i 1 λ F i 1 λ {α ii } i with a weight λ Z n 1 and suitable shifts and relations like i λ = j i λ and j i λ = j i λ, if i j. j Daniel Tubbenhauer Foamation and the HM-basis October / 43

38 sl 3 -foamation We define a 2-functor called foamation, in the following way. Ψ: U(sl d ) W (p) (3 l ) On objects: The functor is defined by sending an sl d -weight λ = (λ 1,...,λ d 1 ) to an object Ψ(λ) of W (p) (3 l ) by Ψ(λ) = S, S = (a 1,...,a l ), a i {0,1,2,3}, λ i = a i+1 a i, l a i = 3 l. i=1 On morphisms: The functor on morphisms is by glueing the ladder webs from before on top of the sl 3 -webs in W (3l ). Daniel Tubbenhauer Foamation and the HM-basis October / 43

39 sl 3 -foamation On 2-cells: We define i,λ λ i λ i+1 i,λ λ i λ i+1 i,i,λ λ i λ i+1 i,i+1,λ ( 1) λi+1 λ i λ i+1 λ i+2 i+1,i,λ λ i λ i+1 λ i+2 And some others. Daniel Tubbenhauer Foamation and the HM-basis October / 43

40 HM-basis of the cyclotomic quotient The category U(sl d ) has a certain subquotient, called the cyclotomic Khovanov-Lauda and Rouquier (KL-R) algebra R Λ, that categorifies the irreducible U q (sl d )-representation of highest weight Λ. It is defined by taking only downwards pointing arrow s (only F s) and mod out by the so-called cyclotomic relation λ i 1 i 2 i 3 i n λ n Based on work of Brundan, Kleshchev and Wang, Hu and Mathas have defined a graded cellular basis for R Λ. We also have only F s - could this be a coincidence? Daniel Tubbenhauer Foamation and the HM-basis October / 43

41 The growth algorithm for foams gives the HM-basis Theorem(T) The growth algorithm for foams gives the HM-basis under categorified q-skew Howe duality. Corollary( Almost directly) We have ψ p ([D λ p ]) = bλ under the isometry ψ p : K 0 (Wp (3 l ) ) Z[q,q 1 ] Q(q) W (3 l ), that is projective covers Dp λ (who give a complete list of all projective, irreducible K S -modules) of the simple heads D λ of the cell modules C λ categorify the upper global crystal basis b λ of W (3l ). In principle, the Dp λ are computable from the extended growth algorithm. Daniel Tubbenhauer Foamation and the HM-basis October / 43

42 Exempli gratia Every web has a graded cellular basis parametrised by flow lines. wt=3 wt= 1 q deg=0 q deg=4 wt=1 wt= 1 q deg=2 q deg=4 wt=1 wt= 3 q deg=2 q deg=6 That these foams are really a graded cellular basis follows from our theorem. Note that the Kuperberg bracket gives [2][3] = q 3 +2q 1 +2q +q 3. Daniel Tubbenhauer Foamation and the HM-basis October / 43

43 There is still much to do... Daniel Tubbenhauer Foamation and the HM-basis October / 43

44 Thanks for your attention! Daniel Tubbenhauer Foamation and the HM-basis October / 43

sl 3 -web bases, categorification and link invariants

sl 3 -web bases, categorification and link invariants Daniel Tubbenhauer Joint work with Marco Mackaay and Weiwei Pan November 2013 F 1 1 1 2 2 F (2) 2 2 0 2 2 F 1 2 2 0 2 F (2) 3 F (2) 2 3 1 0 2 3 1 2