HIGHEST WEIGHT CATEGORIES ARISING FROM KHOVANOV S DIAGRAM ALGEBRA IV: THE GENERAL LINEAR SUPERGROUP

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1 HIGHEST WEIGHT CATEGORIES ARISING FROM KHOVANOV S DIAGRAM ALGEBRA IV: THE GENERAL LINEAR SUPERGROUP JONATHAN BRUNDAN AND CATHARINA STROPPEL Abstract. We prove that blocks of the general linear supergroup are Morita equivalent to a limiting version of Khovanov s iagram algebra. We euce that blocks of the general linear supergroup are Koszul. Contents 1. Introuction 1 2. Combinatorics of Grothenieck groups 8 3. Cyclotomic Hecke algebras an level two Schur-Weyl uality Morita equivalence with generalise Khovanov algebras Direct limits 35 Inex of notation 43 References Introuction This is the culmination of a series of four articles stuying various generalisations of Khovanov s iagram algebra from [Kh]. The goal is to relate the limiting version Hr of this algebra constructe in [BS1] to blocks of the general linear supergroup GL(m n). More precisely, working always over a fixe algebraically close fiel F of characteristic zero, we show that any block of GL(m n) of atypicality r is Morita equivalent to the algebra Hr. We refer the reaer to the introuction of [BS1] for a etaile account of our approach to the efinition of Khovanov s iagram algebra an the construction of its limiting version; see also [St3] which iscusses further the connections to link homology. To formulate our main result in etail, fix m, n 0 an let G enote the algebraic supergroup GL(m n) over F. Using scheme-theoretic language, G can be regare as a functor from the category of commutative superalgebras over F to the category of groups, mapping a commutative superalgebra A = A 0 A 1 to the group G(A) of all invertible (m + n) (m + n) matrices of the form ( ) a b g = (1.1) c 2010 Mathematics Subject Classification: 17B10, 16S37. First author supporte in part by NSF grant no. DMS

2 2 JONATHAN BRUNDAN AND CATHARINA STROPPEL where a (resp. ) is an m m (resp. n n) matrix with entries in A 0, an b (resp. c) is an m n (resp. n m) matrix with entries in A 1. We are intereste here in finite imensional representations of G, which can be viewe equivalently as integrable supermoules over its Lie superalgebra g = gl(m n, F); the conition for integrability is the same as for g 0 = gl(m, F) gl(n, F). For example, we have the natural G-moule V of column vectors, with stanar basis v 1,..., v m, v m+1,..., v m+n an Z 2 -graing efine by putting v r in egree r := 0 if 1 r m, r := 1 if m + 1 r m + n. Let B an T be the stanar choices of Borel subgroup an maximal torus: for each commutative superalgebra A, the groups B(A) an T (A) consist of all matrices g G(A) that are upper triangular an iagonal, respectively. Let ε 1,..., ε m+n be the usual basis for the character group X(T ) of T, i.e. ε r picks out the rth iagonal entry of a iagonal matrix. Equip X(T ) with a symmetric bilinear form (.,.) such that (ε r, ε s ) = ( 1) r δ r,s, an set m n ρ := (1 r)ε r + (m s)ε m+s. (1.2) Let r=1 { X + (T ) := λ X(T ) s=1 (λ + ρ, ε 1 ) > > (λ + ρ, ε m ), (λ + ρ, ε m+1 ) < < (λ + ρ, ε m+n ) } (1.3) enote the set of ominant weights. We allow only even morphisms between G-moules, so that the category of all finite imensional G-moules is obviously an abelian category. Any G-moule M ecomposes as M = M + M, where M + (resp. M ) is the G-submoule of M spanne by the egree λ (resp. the egree ( λ + 1)) component of the λ- weight space of M for all λ X(T ); here λ := (λ, ε m ε m+n ) (mo 2). It follows that the category of all finite imensional G-moules ecomposes as F ΠF, where F = F(m n) (resp. ΠF = ΠF(m n)) is the full subcategory consisting of all M such that M = M + (resp. M = M ). Moreover F an ΠF are obviously equivalent. In view of this ecomposition, we will focus just on F from now on. Note further that F is close uner tensor prouct, an it contains both the natural moule V an its ual V. By [B1, Theorem 4.47], the category F is a highest weight category with weight poset (X + (T ), ), where is the Bruhat orering efine combinatorially in the next paragraph. We fix representatives {L(λ) λ X + (T )} for the isomorphism classes of irreucible moules in F so that L(λ) is an irreucible object in F generate by a one-imensional B-submoule of weight λ. We also enote the stanar an projective inecomposable moules in the highest weight category F by {V(λ) λ X + (T )} an {P(λ) λ X + (T )}, respectively. So P(λ) V(λ) L(λ). In this setting, the stanar moule V(λ) is often referre to as a Kac moule after [Ka]. Now we turn our attention to the iagram algebra sie. Let Λ = Λ(m n) enote the set of all weights in the iagrammatic sense of [BS1, 2] rawn on a number line with vertices inexe by Z, such that a total of m vertices are labelle or, a total of n vertices are labelle or, an all of the (infinitely many) remaining vertices are labelle. From now on, we ientify the set

3 KHOVANOV S DIAGRAM ALGEBRA IV 3 X + (T ) introuce above with the set Λ via the following weight ictionary. Given λ X + (T ), we efine I (λ) := {(λ + ρ, ε 1 ),..., (λ + ρ, ε m )}, (1.4) I (λ) := Z \ {(λ + ρ, ε m+1 ),..., (λ + ρ, ε m+n )}. (1.5) Then we ientify λ with the element of Λ whose ith vertex is labelle if i oes not belong to either I (λ) or I (λ), if i belongs to I (λ) but not to I (λ), if i belongs to I (λ) but not to I (λ), if i belongs to both I (λ) an I (λ). (1.6) For example, the zero weight (which parametrises the trivial G-moule) is ientifie with the iagram n m n {}}{{}}{ if m n, }{{}}{{} m n m if m n. where the leftmost is on vertex (1 m). In these iagrammatic terms, the Bruhat orering on X + (T ) mentione earlier is the same as the Bruhat orering on Λ from [BS1, 2], that is, the partial orer on iagrams generate by the basic operation of swapping a an an so that s move to the right. Let be the equivalence relation on Λ generate by permuting s an s. Following the language of [BS1] again, the -equivalence classes of weights from Λ are calle blocks. The efect ef(γ) of each block Γ Λ/ is simply equal to the number of vertices labelle in any weight λ Γ; this is the same thing as the usual notion of atypicality in the representation theory of GL(m n) as in e.g. [Se1, (1.1)]. Let K = K(m n) enote the irect sum of the iagram algebras K Γ associate to all the blocks Γ Λ/ as efine in [BS1, 4]. As a vector space, K has a basis {(aλb) for all oriente circle iagrams aλb with λ Λ}, (1.7) an its multiplication is efine by an explicit combinatorial proceure in terms of such iagrams as in [BS1, 6]; see the iscussion at the en of this introuction for some examples illustrating the precise meaning of all this. As explaine in [BS1, 5], to each λ Λ there is associate an iempotent e λ K. The left ieal P (λ) := Ke λ is a projective inecomposable moule with irreucible hea enote L(λ). The moules {L(λ) λ Λ} are all one imensional an give a complete set of irreucible K-moules. Finally let V (λ) be the stanar moule corresponing to λ, which was referre to as a cell moule in [BS1, 5]. The main result of the paper is the following. Theorem 1.1. There is an equivalence of categories E from F(m n) to the category of finite imensional left K(m n)-moules, such that EL(λ) = L(λ), EV(λ) = V (λ) an EP(λ) = P (λ) for each λ Λ(m n).

4 4 JONATHAN BRUNDAN AND CATHARINA STROPPEL Our proof of Theorem 1.1 involves showing that K is isomorphic to the locally finite enomorphism algebra En fin G (P )op of a canonical minimal projective generator P = λ Λ P(λ) for F; see Lemmas below. To construct P, we first consier the weight m n λ p,q := pε r (q + m)ε m+s (1.8) r=1 s=1 for integers p q. This is represente iagrammatically by p m p q } {{ }} {{ } (1.9) m where the rightmost is on vertex p an the rightmost is on vertex (q + n). The G-moule V(λ p,q ) is projective, hence the tensor space V(λ p,q ) V is projective for any 0. Moreover, any P(λ) appears as a summan of V(λ p,q ) V for suitable p, q an. The key step in our approach is to compute the enomorphism algebra of V(λ p,q ) V for 0. For min(m, n), we show that it is a certain egenerate cyclotomic Hecke algebra of level two, giving a new super version of the level two Schur-Weyl uality from [BK1]. Then we invoke results from [BS3] which show that the basic algebra that is Morita equivalent to this cyclotomic Hecke algebra is a generalise Khovanov algebra; this equivalence relies in particular on the connection between cyclotomic Hecke algebras an Khovanov-Laua-Rouquier algebras in type A from [BK2]. Finally we let p, q an vary, taking a suitable irect limit to erive our main result. We briefly collect here some applications of Theorem 1.1. Blocks of the same atypicality are equivalent. The algebras K Γ for all Γ Λ(m n)/ are the blocks of the algebra K(m n). Hence by Theorem 1.1 they are the basic algebras representing the iniviual blocks of the category F(m n). In the iagrammatic setting, it is obvious for Γ Λ(m n)/ an Γ Λ(m n )/ (for possibly ifferent m an n ) that the algebras K Γ an K Γ are isomorphic if an only if Γ an Γ have the same efect. Thus we recover a result of Serganova from [Se2]: the blocks of GL(m n) for all m, n epen up to equivalence only on the egree of atypicality of the block. Graings on blocks an Koszulity. Each of the algebras K Γ carries a canonical positive graing with respect to which it is a (locally unital) Koszul algebra; see [BS2, Corollary 5.13]. So Theorem 1.1 implies that blocks of GL(m n) are Koszul. The appearence of such hien Koszul graings in representation theory goes back to the classic paper of Beilinson, Ginzburg an Soergel [BGS] on blocks of category O for a semisimple Lie algebra. In that work, the graing is of geometric origin, whereas in our situation we establish the Koszulity in a purely algebraic way. Rigiity of Kac moules. Another consequence of Theorem 1.1, combine with [BS2, Corollary 6.7] on the iagram algebra sie, is that all the Kac moules V(λ) are rigi, i.e. their raical an socle filtrations coincie. See [BS1, Theorem 5.2] for the explicit combinatorial escription of the layers. n q+n

5 KHOVANOV S DIAGRAM ALGEBRA IV 5 Kostant moules an BGG resolution. In [BS2] we stuie in etail the Kostant moules for the generalise Khovanov algebras, i.e. the irreucible moules whose Kazhan-Lusztig polynomials are multiplicity-free. In particular in [BS2, Lemma 7.2] we classifie the highest weights of these moules via a pattern avoiance conition. Combining this with Theorem 1.1, we obtain the following classification of all Kostant moules for GL(m n): they are the irreucible moules parametrise by the weights in which no two vertices labelle have a vertex labelle between them. By [BS2, Theorem 7.3], Kostant moules possess a BGG resolution by multiplicity-free irect sums of stanar moules. All irreucible polynomial representations of GL(m n) satisfy the combinatorial criterion to be Kostant moules, so this gives another proof of the main result of [CKL]. Enomorphism algebras of PIMs. For any λ Λ, Theorem 1.1 implies that the enomorphism algebra En G (P(λ)) op of the projective inecomposable moule P(λ) is isomorphic to the algebra e λ Ke λ. By the efinition of multiplication in K, this algebra is isomorphic to F[x 1,..., x r ]/(x 2 1,..., x2 r) where r is the efect (atypicality) of the block containing λ, answering a question raise recently by several authors; see [BKN, (4.2)] an [Dr, Conjecture 4.3.3]. (It shoul also be possible to give a proof of the commutativity of these enomorphism algebras using some eformation theory like in [St2, 2.8], invoking the fact that the multiplicities (P(λ) : V(µ)) are at most one by Theorem 2.1 below; see [St1, Theorem 7.1] for a similar situation.) Super uality. When combine with the results from [BS3], our results can be use to prove the Super Duality Conjecture as formulate in [CWZ]. A irect algebraic proof of this conjecture, an its substantial generalisation from [CW], has recently been foun by Cheng an Lam [CL]. All of these results suggest some more irect geometric connection between the representation theory of GL(m n) an the category of perverse sheaves on Grassmannians may exist. To conclue this introuction, we recall in more etail the efinition of the algebra K following [BS1, 6]. We assume m = n = r an focus just on the principal block of G = GL(r r), which is the basic example of a block of atypicality r. The ominant weights in this block are the weights r λ i1,...,i r := (i s + s 1)(ε s ε 2r+1 s ) X + (T ) s=1 parametrise by sequences i 1 > > i r of integers. Accoring to the weight ictionary (1.6), the iagram for λ i1,...,i r has label at the vertices inexe by i 1,..., i r, an label at all remaining vertices. The corresponing block of the algebra K is exactly the algebra enote Kr in the introuction of [BS1]. Theorem 1.1 (or rather the more precise Lemmas below) asserts in this situation that ( ) op Kr = En fin G P(λ i1,...,i r ). (1.10) i 1 > >i r Our explicit basis of Kr is given by the oriente circle iagrams from (1.7). These are obtaine by taking the iagram of some λ i1,...,i r, then gluing r cups

6 6 JONATHAN BRUNDAN AND CATHARINA STROPPEL an infinitely many rays to the bottom an r caps an infinitely many rays to the top of the iagram so that every vertex meets exactly one cup or ray below an exactly one cap or ray above the number line; each cup an each cap is incient with one vertex labelle an one vertex labelle ; each ray is incient with a vertex labelle an extens from there vertically up or own to infinity; no crossings of cups, caps an rays are allowe. Uner the isomorphism (1.10), such a iagram represents a homomorphism P(λ j1,...,j r ) P(λ k1,...,k r ) where j 1 > > j r (resp. k 1 > > k r ) inex the leftmost vertices of the cups (resp. caps) in the iagram. For example, here are two oriente circle iagrams corresponing to basis vectors in K 2 (where inicates infinitely many pairs of vertical rays labelle ): The first iagram here represents a homomorphism P(λ 4,2 ) P(λ 3,2 ) an the secon one represents a homomorphism P(λ 3,2 ) P(λ 4,1 ). Multiplying these two basis vectors together as escribe in the next paragraph, bearing in min the op in (1.10) which means for once that we are writing maps on the right, one gets the basis vector which is some homomorphism P(λ 4,2 ) P(λ 4,1 ). We now sketch briefly the combinatorial proceure for multiplying basis vectors. Given two basis vectors, their prouct is necessarily zero unless the caps at the top of the first iagram are in exactly the same positions as the cups at the bottom of the secon. Assuming that is the case, we glue the first iagram unerneath the secon an join matching pairs of rays. Then we perform a sequence of generalise surgery proceures to smooth out all cup-cap pairs in the symmetric mile section of the resulting composite iagram, obtaining zero or more new iagrams in which the mile section only involves vertical line segments. Finally we collapse these mile sections to obtain a sum of basis vectors, which is the esire prouct. Each generalise surgery proceure in this algorithm either involves two components in the iagram merging into one or one component splitting into two. The rules for relabelling the new component(s) prouce when this operation is performe are summarize as

7 follows: KHOVANOV S DIAGRAM ALGEBRA IV , 1 x x, x x 0, 1 y y, x y 0, y y 0, 1 1 x + x 1, x x x, y x y, where 1 represents an anti-clockwise circle, x represents a clockwise circle, an y represents a line. This is a little cryptic; we refer the reaer to [BS1] for a fuller account (an explanation of the connection to Khovanov s original construction via a certain TQFT). Let us at least apply this algorithm to the example from the previous paragraph: two surgeries are neee, the first of which involves an anti-clockwise circle an a line merging together (1 y y) an the secon of which involves a line splitting into a clockwise circle an a line (y x y): Contracting the mile section of the iagram on the right han sie here gives the final prouct recore alreay at the en of the previous paragraph. The case r = 1 in the above iscussion (the principal block of GL(1 1)) is easy to erive from scratch, but still this is quite instructive. So now the irreucible moules are inexe simply by the weights {λ i i Z}. It is well known that P(λ i ) has irreucible socle an hea isomorphic to L(λ i ), with ra P(λ i )/soc P(λ i ) = L(λ i 1 ) L(λ i+1 ). Hence in this case the locally finite enomorphism algebra from (1.10) has basis {e i, c i, a i, b i i Z}, where e i is the projection onto P(λ i ), a i : P(λ i ) P(λ i+1 ) an b i : P(λ i+1 ) P(λ i ) are non-zero homomorphisms chosen so that b i a i = a i 1 b i 1, an c i := b i a i sens the hea of P(λ i ) onto its socle. This correspons to our iagram basis for K1 so that e i = c i = a i = b i = where we isplay only vertices i, i + 1, i + 2 an there are infinitely many pairs of vertical rays labelle at all other vertices. In fact, K1 is simply the path algebra of the infinite quiver a i 1 b i 1 a i b i a i+1 b i+1 a i+2 b i+2 moulo the relations a i b i = b i 1 a i 1 an a i a i+1 = 0 = b i+1 b i for all i Z. It is clear from the quiver escription that K1 is naturally grae by path length; this is actually a Koszul graing. For general r the canonical Koszul graing on Kr is efine by eclaring that a basis vector is of egree equal to the total number of clockwise cups an caps in the oriente circle iagram.

8 8 JONATHAN BRUNDAN AND CATHARINA STROPPEL Acknowlegements. This article was written up uring stays by both authors at the Isaac Newton Institute in Spring We thank the INI staff an the Algebraic Lie Theory programme organisers for the opportunity. 2. Combinatorics of Grothenieck groups In this preliminary section, we compare the combinatorics unerlying the representation theory of GL(m n) with that of the iagram algebra K(m n). Our exposition is largely inepenent of [B1], inee, we will reprove the relevant results from there as we go. On the other han, we o assume that the reaer is familiar with the general theory of iagram algebras evelope in [BS1, BS2]. Later in the article we will also nee to appeal to various results from [BS3]. Representation theory of K(m n). Fix once an for all integers m, n 0. Let K = K(m n) an Λ = Λ(m n) be as in the introuction. The elements {e λ λ Λ} form a system of (in general infinitely many) mutually orthogonal iempotents in K such that K = e λ Ke µ. (2.1) λ,µ Λ So the algebra K is locally unital, but it is not unital (except in the trivial case m = n = 0). By a K-moule we always mean a locally unital moule; for a left K-moule M this means that M ecomposes as M = λ Λ e λ M. The irreucible K-moules {L(λ) λ Λ} efine in the introuction are all one imensional, so K is a basic algebra. Let rep(k) enote the category of finite imensional left K-moules. The Grothenieck group [rep(k)] of this category is the free Z-moule on basis {[L(λ)] λ Λ}. The stanar moules {V (λ) λ Λ} an the projective inecomposable moules {P (λ) λ Λ} from [BS1, 5] are finite imensional, so it makes sense to consier their classes [V (λ)] an [P (λ)] in [rep(k)]. Finally, we use the notation µ λ (resp. µ λ) from [BS1, 2] to inicate that the composite iagram µλ (resp. µλ) is oriente in the obvious sense. Theorem 2.1. We have in [rep(k)] that for each λ Λ. [P (λ)] = µ λ [V (µ)], [V (λ)] = µ λ[l(µ)] Proof. This follows from [BS1, Theorem 5.1] an [BS1, Theorem 5.2]. As µ λ (resp. µ λ) implies that µ λ (resp. µ λ) in the Bruhat orering, we euce from Theorem 2.1 that the classes {[P (λ)]} an {[V (λ)]} are linearly inepenent in [rep(k)]. However they o not span [rep(k)] as the chains in the Bruhat orer are infinite.

9 KHOVANOV S DIAGRAM ALGEBRA IV 9 Remark 2.2. The algebra K possesses a natural Z-graing efine by eclaring that each basis vector (aλb) from (1.7) is of egree equal to the number of clockwise cups an caps in the iagram aλb. This means that one can consier the grae representation theory of K. The various moules L(λ), V (λ) an P (λ) also possess canonical graings, as is iscusse in etail in [BS1, 5]. Special projective functors: the iagram sie. As in [BS3, (2.5)], let us represent a block Γ Λ/ by means of its block iagram, that is, the iagram obtaine by taking any λ Γ an replacing all the s an s labelling its vertices by the symbol. Because m an n are fixe, the block Γ can be recovere uniquely from its block iagram. Recall also the notion of the efect of a weight λ Λ from [BS1, 2]. In this setting, this simply means the number of vertices labelle in λ, an the efect of λ is the same thing as the efect ef(γ) of the unique block Γ Λ/ containing λ. Given a block Γ, we say that i Z is Γ-amissible if the ith an (i + 1)th vertices of the block iagram of Γ match the top number line of a unique one of the following pictures, an ef(γ) is as inicate: F i ef(γ) 1 ef(γ) 0 ef(γ) 0 ef(γ) 0 cup cap right-shift left-shift Γ t i (Γ) Γ α i (2.2) Assuming i is Γ-amissible, we let (Γ α i ) enote the block obtaine from Γ by relabelling the ith an (i + 1)th vertices of its block iagram accoring to the bottom number line of the appropriate picture. Also efine a (Γ α i )Γ-matching t i (Γ) in the sense of [BS2, 2] so that the strip between the ith an (i + 1)th vertices of t i (Γ) is as in the picture, an there are only vertical ientity line segments elsewhere. For blocks Γ, Λ/ an a Γ -matching t, recall the geometric bimoule KΓ t from [BS2, 3]. By efinition this is a (K Γ, K )-bimoule. We can view it as a (K, K)-bimoule by extening the actions of K Γ an K to all of K so that the other blocks act as zero. The functor KΓ t K? is an enofunctor of rep(k) calle a projective functor. Writing t for the mirror image of t in a horizontal axis, the functor K Γ t K? gives another projective functor which is biajoint to KΓ t K? by [BS2, Corollary 4.9]. For any i Z, introuce the (K, K)-bimoules F i := K t i(γ) (Γ α i )Γ, Ẽ i := K t i(γ) Γ(Γ α i ), (2.3) Γ Γ where the irect sums are over all Γ Λ/ such that i is Γ-amissible. The special projective functors are the enofunctors F i := F i K? an E i := Ẽi K? of rep(k) efine by tensoring with these bimoules. The iscussion in the previous paragraph implies that the functors F i an E i are biajoint, hence they are both exact an map projectives to projectives. For λ Λ, let I (λ) := I (λ) I (λ) (resp. I (λ) := Z \ (I (λ) I (λ))) enote the set of integers inexing the vertices labelle (resp. ) in λ; cf.

10 10 JONATHAN BRUNDAN AND CATHARINA STROPPEL (1.6). Introuce the notion of the height of λ: ht(λ) := i i I (λ) i I (λ) i. (2.4) Note all weights belonging to the same block have the same height. Lemma 2.3. For λ Λ an i Z, all composition factors of F i L(λ) (resp. E i L(λ)) are of the form L(µ) with ht(µ) = ht(λ) + 1 (resp. ht(λ) 1). Proof. This follows by inspecting (2.2). Lemma 2.4. Let λ Λ an i Z. For symbols x, y {,,, } we write λ xy for the iagram obtaine from λ by relabelling the ith an (i + 1)th vertices by x an y, respectively. (i) If λ = λ then F i P (λ) = P (λ ), F i V (λ) = V (λ ), F i L(λ) = L(λ ). (ii) If λ = λ then F i P (λ) = P (λ ), F i V (λ) = V (λ ), F i L(λ) = L(λ ). (iii) If λ = λ then F i P (λ) = P (λ ), F i V (λ) = V (λ ), F i L(λ) = L(λ ). (iv) If λ = λ then F i P (λ) = P (λ ), F i V (λ) = V (λ ), F i L(λ) = L(λ ). (v) If λ = λ then: (a) F i P (λ) = P (λ ); (b) there is a short exact sequence 0 V (λ ) F i V (λ) V (λ ) 0; (c) F i L(λ) has irreucible socle an hea both isomorphic to L(λ ), an all other composition factors are of the form L(µ) for µ Λ such that µ = µ, µ = µ or µ = µ. (vi) If λ = λ then F i P (λ) = P (λ ) P (λ ), F i V (λ) = V (λ ) an F i L(λ) = L(λ ). (vii) If λ = λ then F i V (λ) = V (λ ) an F i L(λ) = {0}. (viii) If λ = λ then F i V (λ) = F i L(λ) = {0}. (ix) If λ = λ then F i V (λ) = F i L(λ) = {0}. (x) For all other λ we have that F i P (λ) = F i V (λ) = F i L(λ) = {0}. For the ual statement about E i, interchange all occurrences of an. Proof. Apply [BS2, Theorems 4.2], [BS2, Theorem 4.5] an [BS2, Theorem 4.11], exactly as was one in [BS3, Lemma 3.4]. Remark 2.5. Using Lemma 2.4, one can check that the enomorphisms of [rep(k)] inuce by the functors F i an E i for all i Z satisfy the Serre relations efining the Lie algebra sl. Inee, letting V enote the natural sl -moule of column vectors, the category rep(k) can be interprete in a precise sense as a categorification of a certain completion of the sl -moule m V n V ; see also [B1] where this point of view is taken on the supergroup sie. Using the grae representation theory mentione in Remark 2.2, i.e. replacing rep(k) with the category of finite imensional grae K-moules, one gets a categorification of the q-analogue of this moule over the quantise enveloping algebra U q (sl ); the action of q comes from shifting the graing on a moule up by one. We are not going to pursue this connection further here, but refer the reaer to [BS3, Theorem 3.5] where an analogous grae categorification theorem is iscusse in etail.

11 KHOVANOV S DIAGRAM ALGEBRA IV 11 The crystal graph. Define the crystal graph to be the irecte coloure graph with vertex set equal to Λ an a irecte ege µ i λ of colour i Z if L(λ) is a quotient of F i L(µ). It is clear from Lemma 2.4 that µ i λ if an only if the ith an (i + 1)th vertices of λ an µ are labelle accoring to one of the six cases in the following table, an all other vertices of λ an µ are labelle in the same way: µ λ (2.5) Comparing this explicit escription with [B1, 3-], it follows that our crystal graph is isomorphic to Kashiwara s crystal graph associate to the sl -moule mentione in Remark 2.5, which hopefully explains our choice of terminology. Suppose we are given integers p q. Define the following intervals I p,q := {p m + 1, p m + 2,..., q + n 1}, (2.6) I + p,q := {p m + 1, p m + 2,..., q + n 1, q + n}. (2.7) (The reaer may fin it helpful at this point to note which vertices of the weight λ p,q from (1.9) are inexe by the set I + p,q.) Then introuce the following subsets of Λ: Λ p,q := {λ Λ the ith vertex of λ is labelle for all i / I p,q}, + (2.8) { Λ amongst vertices j,..., q + n of λ, the number } p,q := λ Λ p,q of s is the number of s, for all j I p,q +. (2.9) Note that the weight λ p,q from (1.9) belongs to Λ p,q. It is the unique weight in Λ p,q of minimal height. Lemma 2.6. Given λ Λ, choose p q such that λ Λ p,q (which is always possible as there are infinitely many s an finitely many s). Then there are i integers i 1,..., i I p,q, where = ht(λ) ht(λ p,q ), such that λ 1 i p,q λ is a path in the crystal graph. Moreover we have that F i F i1 V (λ p,q ) = P (λ) 2r, where r is the number of eges in the given path of the form. Proof. For the first statement, we procee by inuction on ht(λ). If ht(λ) = ht(λ p,q ), then λ = λ p,q an the conclusion is trivial. Now assume that ht(λ) > ht(λ p,q ). As λ Λ p,q an λ λ p,q, it is possible to fin i I p,q such that the ith an (i + 1)th vertices of λ are labelle,,,, or. Inspecting (2.5), there is a unique weight µ Λ p,q with µ i λ in the crystal graph. Noting ht(µ) = ht(λ) 1, we are now one by inuction. To euce the secon statement, we apply Lemma 2.4 to get easily that F i F i1 P (λ p,q ) = P (λ) 2r. Finally P (λ p,q ) = V (λ p,q ) as λ p,q is of efect zero, by [BS1, Theorem 5.1]. Representation theory of GL(m n). Now we turn to iscussing the representation theory of G = GL(m n). In the introuction, we efine alreay the abelian category F = F(m n) an the irreucible moules {L(λ)}, the stanar moules {V(λ)} an the projective inecomposable moules {P(λ)}, all of

12 12 JONATHAN BRUNDAN AND CATHARINA STROPPEL which are parametrise by the set X + (T ) of ominant weights. We are using an unusual font here (an a few other places later on) to avoi confusion with the analogous K-moules {L(λ)}, {V (λ)} an {P (λ)}. Recall in particular that the Z 2 -graing on L(λ) is efine so that its λ-weight space is concentrate in egree λ := (λ, ε m ε m+n ) (mo 2). Bearing in min that we consier only even morphisms, the moules {L(λ) λ X + (T )} {ΠL(λ) λ X + (T )} give a complete set of pairwise non-isomorphic irreucible G-moules, where Π enotes the change of parity functor. The stanar moule V(λ) is usually calle a Kac moule in this setting after [Ka], an can be constructe explicitly as follows. Let P be the parabolic subgroup of G such that P (A) consists of all invertible matrices of the form (1.1) with c = 0, for each commutative superalgebra A. Given λ X + (T ), we let E(λ) enote the usual finite imensional irreucible moule of highest weight λ for the unerlying even subgroup G 0 = GL(m) GL(n), viewing E(λ) as a supermoule with Z 2 -graing concentrate in egree λ. We can regar E(λ) also as a P -moule by inflating through the obvious homomorphism P G 0. Then we have that V(λ) = U(g) U(p) E(λ), (2.10) where g an p enote the Lie superalgebras of G an P, respectively. This construction makes sense because the inuce moule on the right han sie of (2.10) is an integrable g-supermoule, i.e. it lifts in a unique way to a G-moule. The moule L(λ) is isomorphic to the unique irreucible quotient of V(λ). Also P(λ) is the projective cover of L(λ) in the category F. It has a stanar flag, that is, a filtration whose sections are stanar moules. The multiplicity (P(λ) : V(µ)) of V(µ) as a section of any such stanar flag is given by the BGG reciprocity formula (P(λ) : V(µ)) = [V(µ) : L(λ)], (2.11) as follows from [Zo] or the iscussion in [B2, Example 7.5]. Special projective functors: the supergroup sie. Recall the weight ictionary from (1.6) by means of which we ientify the set X + (T ) with the set Λ = Λ(m n). Uner this ientification, the usual notion of the egree of atypicality of a weight λ X + (T ) correspons to the notion of efect of λ Λ. Given λ, µ Λ, the irreucible G-moules L(λ) an L(µ) have the same central character if an only if λ µ in the iagrammatic sense; this can be euce from [Se1, Corollary 1.9]. Hence the category F ecomposes as F = F Γ, (2.12) Γ Λ/ where F Γ is the full subcategory consisting of the moules all of whose composition factors are of the form L(λ) for λ Γ. We let pr Γ : F F be the exact functor efine by projection onto F Γ along (2.12). Recall that V enotes the natural G-moule an V is its ual. Following [B1, (4.21) (4.22)], we efine the special projective functors F i an E i for each

13 KHOVANOV S DIAGRAM ALGEBRA IV 13 i Z to be the following enofunctors of F: F i := pr Γ αi (? V ) pr Γ, E i := Γ Γ pr Γ (? V ) pr Γ αi, (2.13) where the irect sums are over all Γ Λ/ such that i is Γ-amissible (as in (2.3)). The functors? V an? V are biajoint, hence so are F i an E i. In particular, all these functors are exact an sen projectives to projectives. For later use, let us fix once an for all a choice of an ajunction making (F i, E i ) into an ajoint pair for each i Z. Lemma 2.7. The following hol for any λ X + (T ): (i) V(λ) V has a filtration with sections V(λ + ε r ) for all r = 1,..., m + n such that λ + ε r X + (T ), arrange in orer from bottom to top. (ii) V(λ) V has a filtration with sections V(λ ε r ) for all r = 1,..., m+n such that λ ε r X + (T ), arrange in orer from top to bottom. Proof. This follows from the efinition (2.10) an the tensor ientity. Corollary 2.8. The following hol for any λ X + (T ) an i Z: (i) F i V(λ) has a filtration with sections V(λ + ε r ) for all r = 1,..., m + n such that λ + ε r X + (T ) an (λ + ρ, ε r ) = i + (1 ( 1) r )/2, arrange in orer from bottom to top. (ii) E i V(λ) has a filtration with sections V(λ ε r ) for all r = 1,..., m + n such that λ ε r X + (T ) an (λ + ρ, ε r ) = i + (1 + ( 1) r )/2, arrange in orer from top to bottom. Proof. For (i), apply pr Γ αi to the statement of Lemma 2.7(i), where Γ is the block containing λ (an o a little work to translate the combinatorics). The proof of (ii) is similar. Corollary 2.9. We have that? V = i Z F i an? V = i Z E i. The next lemma gives an alternative efinition of the functors F i an E i which will be neee in the next section; cf. [CW, Proposition 5.2]. Let m+n Ω := ( 1) s e r,s e s,r g g, (2.14) r,s=1 where e r,s enotes the rs-matrix unit. This correspons to the supertrace form on g, so left multiplication by Ω (interprete with the usual superalgebra sign conventions) efines a G-moule enomorphism of M N for any G-moules M an N. Lemma For any G-moule M, we have that F i M (resp. E i M) is the generalise i-eigenspace (resp. the generalise (m n + i)-eigenspace) of the operator Ω acting on M V (resp. M V ). Proof. We just explain for F i. Let c := m+n r,s=1 ( 1) s e r,s e s,r U(g) be the Casimir element. It acts on V(λ) by multiplication by the scalar c λ := (λ + 2ρ + (m n 1)δ, λ)

14 14 JONATHAN BRUNDAN AND CATHARINA STROPPEL where δ = ε ε m ε m+1 ε m+n. Also, we have that Ω = ( (c) c 1 1 c)/2 where is the comultiplication of U(g). Now to prove the lemma, it suffices to verify it for the special case M = V(λ). Using the observations just mae, we see that multiplication by Ω preserves the filtration from Lemma 2.7(i), an the inuce action of Ω on the section V(λ + ε r ) is by multiplication by the scalar (c λ+εr c λ m + n)/2 = (λ + ρ, ε r ) + (1 ( 1)ī)/2. The result follows on comparing with Corollary 2.8(i). The next two lemmas are the key to unerstaning the representation theory of GL(m n) from a combinatorial point of view. Lemma Let i Z an λ Λ be a weight such that the ith an (i + 1)th vertices of λ are labelle an, respectively. Let µ be the weight obtaine from λ by interchanging the labels on these two vertices. Then L(µ) is a composition factor of V(λ). Proof. This is a reformulation of [Se1, Theorem 5.5]. It can be prove irectly by an explicit calculation with certain lowering operators in U(g) as in [BS3, Lemma 4.8]. Lemma Exactly the same statement as Lemma 2.4 hols in the category F, replacing L(λ), V (λ), P (λ), F i an E i by L(λ), V(λ), P(λ), F i an E i, respectively. Proof. The statements involving V(λ) follow from Corollary 2.8. The remaining parts then follow by mimicking the arguments use to prove [BS3, Lemma 4.9], using Lemma 2.11 in place of [BS3, Lemma 4.8]. Corollary Given λ Λ, pick p, q,, i 1,..., i an r as in Lemma 2.6. Then we have that F i F i1 V(λ p,q ) = P(λ) L 2 r. Proof. We note as λ p,q is of efect zero that it is the only weight in its block. Using also (2.11), this implies that P(λ p,q ) = V(λ p,q ). Given this, the corollary follows from Lemma 2.12 in exactly the same way that Lemma 2.6 was euce from Lemma 2.4. Ientification of Grothenieck groups. Consier the Grothenieck group [F] of F. It is the free Z-moule on basis {[L(λ)] λ Λ}. The exact functors F i an E i (resp. F i an E i ) inuce enomorphisms of the Grothenieck group [F] (resp. [rep(k)]), which we enote by the same notation. The last part of the following theorem recovers the main result of [B1]. Theorem Define a Z-moule isomorphism ι : [F] [rep(k)] by eclaring that ι([l(λ)]) = [L(λ)] for each λ Λ. (i) We have that ι([v(λ)]) = [V (λ)] an ι([p(λ)]) = [P (λ)] for each λ Λ. (ii) For each i Z, we have that F i ι = ι F i an E i ι = ι E i as linear maps from [F] to [rep(k)].

15 KHOVANOV S DIAGRAM ALGEBRA IV 15 (iii) We have in [F] that [P(λ)] = [V(µ)], [V(λ)] = µ λ µ λ[l(µ)] for each λ Λ. Proof. Given λ Λ, let p, q,, r an i 1,..., i be as in Lemma 2.6. Lemma 2.6 an Theorem 2.1, we know alreay that By [P (λ)] = 1 2 r F i F i1 [V (λ p,q )] = µ λ[v (µ)], (2.15) all equalities written in [rep(k)]. In view of Lemma 2.12, the action of F i on the classes of stanar moules in [rep(k)] is escribe by exactly the same matrix as the action of F i on the classes of stanar moules in [F]. So we euce from the secon equality in (2.15) that 1 2 r F i F i1 [V(λ p,q )] = [V(µ)], µ λ equality in [F]. By Corollary 2.13 this also equals [P(λ)], proving the first formula in (iii). The secon formula in (iii) follows from the first an (2.11). Then (i) is immeiate from the efinition of ι an the coincience of the formulae in (iii) an Theorem 2.1. Finally to euce (ii), we have alreay note that ι(f i [V (λ)]) = F i [V(λ)] for every λ. It follows easily from this that ι(f i [P (λ)]) = F i [P(λ)] for every λ. Using also the ajointness of F i an E i (resp. F i an E i ) we euce that [E i L(µ) : L(λ)] = im Hom K (P (λ), E i L(µ)) = im Hom K (F i P (λ), L(µ)) = im Hom G (F i P(λ), L(µ)) = im Hom G (P(λ), E i L(µ)) = [E i L(µ) : L(λ)] for every λ, µ Λ. This is enough to show that ι(e i [L(µ)]) = E i [L(µ)] for every µ, which implies (ii) for E i an E i. The argument for F i an F i is similar. Highest weight structure an uality. At this point, we can also euce the following result, which recovers [B1, Theorem 4.47]. Theorem The category F is a highest weight category in the sense of [CPS] with weight poset (Λ, ). The moules {L(λ)}, {V(λ)} an {P(λ)} give its irreucible, stanar an projective inecomposable moules, respectively. Proof. We alreay note just before (2.11) that P(λ) has a stanar flag with V(λ) at the top. Moreover by Theorem 2.14(iii) all the other sections of this flag are all of the form V(µ) with µ > λ in the Bruhat orer. The theorem follows from this, (2.11) an the efinition of highest weight category. The costanar moules in the highest weight category F can be constructe explicitly as the uals V(λ) of the stanar moules with respect to a natural

16 16 JONATHAN BRUNDAN AND CATHARINA STROPPEL uality. This uality maps a G-moule M to the linear ual M with the action of G efine using the supertranspose anti-automorphism g g st, where ( g st a t c = t ) b t for g of the form (1.1). Note fixes irreucible moules, i.e. L(λ) = L(λ) for each λ Λ. 3. Cyclotomic Hecke algebras an level two Schur-Weyl uality Fix integers p q an let λ p,q be the weight of efect zero from (1.8). The stanar moule V(λ p,q ) is projective. As the functor? V sens projectives to projectives, the G-moule V(λ p,q ) V is again projective for any 0. We want to escribe its enomorphism algebra. Action of the egenerate affine Hecke algebra. We begin by constructing an explicit basis for V(λ p,q ) V. Recalling (2.10), we have that t V(λ p,q ) = U(g) U(p) E(λ p,q ). (3.1) Let et m (resp. et n ) enote the one-imensional G 0-moule efine by taking the eterminant of GL(m) (resp. GL(n)), with Z 2 -graing concentrate in egree 0. Then the moule E(λ p,q ) in (3.1) is the inflation to P of the moule Π n(q+m) (et p m et (q+m) n ), so it is also one imensional. Hence, fixing a non-zero highest weight vector v p,q V(λ p,q ), the inuce moule V(λ p,q ) is of imension 2 mn with basis { m+n m r=m+1 s=1 e τr,s r,s v p,q 0 τ r,s 1 }, (3.2) where the proucts here are taken in any fixe orer (changing the orer only changes the vectors by ±1). Recall also that v 1,..., v m+n is the stanar basis for the natural moule V, from which we get the obvious monomial basis {v i1 v i 1 i 1,..., i m + n} (3.3) for V. Tensoring (3.2) an (3.3), we get the esire basis for V(λ p,q ) V. Now let H be the egenerate affine Hecke algebra from [D]. This is the associative algebra equal as a vector space to F[x 1,..., x ] FS, the tensor prouct of a polynomial algebra an the group algebra of the symmetric group S. Multiplication is efine so that F[x 1,..., x ] F[x 1,..., x ] 1 an FS 1 FS are subalgebras of H, an also s r x s = x s s r if s r, r + 1, s r x r+1 = x r s r + 1, where s r enotes the rth basic transposition (r r + 1). By [CW, Proposition 5.1], there is a right action of H on V(λ p,q ) V by G-moule enomorphisms. The transposition s r acts as the super flip (v v i1 v ir v ir+1 v i )s r = ( 1)īrīr+1 v v i1 v ir+1 v ir v i. This is the same as the enomorphism efine by left multiplication by the element Ω from (2.14) so that the first an secon tensors in Ω hit the (r + 1)th an (r + 2)th tensor positions in V(λ p,q ) V, respectively. The polynomial

17 KHOVANOV S DIAGRAM ALGEBRA IV 17 generator x s acts by left multiplication by Ω so that the first tensor in Ω is sprea across tensor positions 1,..., s using the comultiplication of U(g) an the secon tensor in Ω hits the (s + 1)th tensor position in V(λ p,q ) V. The following lemma gives an explicit formula for the action of x s in a special case. Lemma 3.1. For 1 i 1,..., i m + n an 1 s, we have that (v p,q v i1 v i )x s = pv p,q v i1 v i s 1 + ( 1)īrīs+P r<t<s (ī r +ī s )ī t v p,q v i1 v is v ir v i r=1 if 1 i s m, an (v p,q v i1 v i )x s = (q + m)v p,q v i1 v i s 1 + ( 1)īrīs+P r<t<s (ī r +ī s )ī t v p,q v i1 v is v ir v i r=1 + m ( 1) n(q+m)+ī 1 + +ī s 1 (e is,j v p,q ) v i1 v j v i j=1 if m + 1 i s m + n. (In the first two summations we have interchange v ir an v is, while in the last one we have replace v is by v j.) Proof. Note for any 1 i, j m + n that pv p,q if 1 i = j m, (q + m)v e i,j v p,q = p,q if m + 1 i = j m + n, e i,j v p,q if m + 1 i m + n an 1 j m, 0 otherwise. Using this, the lemma is a routine calculation (taking care with superalgebra signs). Corollary 3.2. The element (x 1 p)(x 1 q) H acts as zero on V(λ p,q ) V. Proof. It suffices to check this in the special case that = 1. In that case, Lemma 3.1 shows that pv p,q v i if 1 i m, (v p,q v i )x 1 = qv p,q v i + m j=1 ( 1)n(q+m) e i,j (v p,q v j ) if m + 1 i m + n. It follows easily that (x 1 p)(x 1 q) acts as zero on the vector v p,q v i for every 1 i m + n. These vectors generate V(λ p,q ) V as a G-moule so we euce that (x 1 p)(x 1 q) acts as zero the whole moule. Corollary 3.3. If min(m, n) then the enomorphisms of V(λ p,q ) V efine by right multiplication by {x σ 1 1 xσ w 0 σ 1,..., σ 1, w S } are linearly inepenent. Proof. Any vector v V(λ p,q ) V can be written as v = i I b i c i where {b i i I} is the basis from (3.2) an the c i s are unique vectors in V. We refer to c i as the b i -component of v. Exploiting the assumption on, we can

18 18 JONATHAN BRUNDAN AND CATHARINA STROPPEL pick istinct integers m + 1 i 1,..., i m + n an 1 j 1,..., j m. Take 0 σ 1,..., σ 1 an consier the vector (v p,q v i1 v i )x σ 1 1 xσ. For 0 τ 1,..., τ 1, Lemma 3.1 implies that the e τ 1 i 1,j 1 e τ i,j v p,q -component of (v p,q v i1 v i )x σ 1 1 xσ is zero either if τ 1 + +τ > σ 1 + +σ, or if τ τ = σ σ but τ r σ r for some r. Moreover, if τ r = σ r for all r, then the e τ 1 i 1,j 1 e τ i,j v p,q -component of (v p,q v i1 v i )x σ 1 1 xσ is equal to ±v k1 v k where k r = i r if σ r = 0 an k r = j r if σ r = 1. This is enough to show that the vectors (v p,q v i1 v i )x σ 1 1 xσ w for all 0 σ 1,..., σ 1 an w S are linearly inepenent, an the corollary follows. In view of Corollary 3.2, the right action of H on V(λ p,q ) V inuces an action of the quotient algebra H p,q := H / (x 1 p)(x 1 q). (3.4) This algebra is a particular example of a egenerate cyclotomic Hecke algebra of level two. It is well known (e.g. see [BK1, Lemma 3.5]) that im H p,q = 2!. Corollary 3.4. If min(m, n) the action of H p,q on V(λ p,q ) V is faithful. Proof. This follows on comparing the imension of H p,q with the number of linearly inepenent enomorphisms constructe in Corollary 3.3. Since the action of H p,q on V(λ p,q ) V is by G-moule enomorphisms, it inuces an algebra homomorphism Φ : H p,q En G (V(λ p,q ) V ) op. (3.5) The main goal in the remainer of the section is to show that this homorphism is surjective. Weight iempotents an the space T p,q. For a tuple i = (i 1,..., i ) Z, there is an iempotent e(i) H p,q etermine uniquely by the property that multiplication by e(i) projects any H p,q -moule onto its i-weight space, that is, the simultaneous generalise eigenspace for the commuting operators x 1,..., x an eigenvalues i 1,..., i, respectively. All but finitely many of the e(i) s are zero, an the non-zero ones give a system of mutually orthogonal iempotents in H p,q summing to 1; see e.g. [BK2, 3.1]. The action of the iempotent e(i) on the moule V(λ p,q ) V can be interprete as follows. In view of Corollary 2.9, we have that V(λ p,q ) V = i Z F i V(λ p,q ) (3.6) where F i enotes the composite F i F i1 of the functors from (2.13). By Lemma 2.10 an the efinition of the actions of x 1,..., x, the summan F i V(λ p,q ) in this ecomposition is precisely the i-weight space of V(λ p,q ) V. Hence the weight iempotent e(i) acts on V(λ p,q ) V as the projection onto the summan F i V(λ p,q ) along the ecomposition (3.6).

19 KHOVANOV S DIAGRAM ALGEBRA IV 19 Recalling the interval I p,q from (2.6), we are usually from now on going to restrict our attention to the summan T p,q := F i V(λ p,q ) (3.7) i (I p,q) of V(λ p,q ) V. By the iscussion in the previous paragraph, we have equivalently that T p,q = (V(λ p,q ) V )1 p,q where 1 p,q :=. (3.8) i (I p,q) e(i) H p,q As a consequence of the fact that any symmetric polynomial in x 1,..., x is central in H, the iempotent 1 p,q is central in H p,q p,q. The space T is naturally a right moule over 1 p,q Hp,q, which is a sum of blocks of Hp,q. Hence the map Φ from (3.5) inuces an algebra homomorphism 1 p,q Hp,q En G (T p,q )op. (3.9) As a refinement of the surjectivity of Φ prove below, we will also see later in the section that the inuce map (3.9) is an isomorphism. Note from (3.16) onwars we will enote the algebra 1 p,q Hp,q instea by R p,q. Stretche iagrams. In this subsection, we evelop some combinatorial tools which will be use initially to compute the imension of the various enomorphism algebras that we are intereste in. We say that a tuple i Z is (p, q)- amissible if i r is Γ r 1 -amissible for each r = 1,...,, where Γ 0,..., Γ are efine recursively from Γ 0 := {λ p,q } an Γ r := Γ r 1 α ir, notation as in (2.2). We refer to the sequence Γ := Γ Γ 1 Γ 0 of blocks here as the associate block sequence. The composite matching t = t t 1 efine by setting t r := t ir (Γ r 1 ) for each r is the associate composite matching. Both of these things make sense only if i Z is (p, q)-amissible. Lemma 3.5. If i Z is not (p, q)-amissible then F i V(λ p,q ) is zero. Proof. This follows from the efinitions an (2.13). By a stretche cap iagram t = t t 1 of height, we mean the associate composite matching for some (p, q)-amissible sequence i Z. We can uniquely recover the sequence i, hence also the associate block sequence Γ, from the stretche cap iagram t. Here is an example of a stretche cap iagram of height 5, taking m = 2, n = 1 an q p = 1; we raw only the strip containing the vertices inexe by I + p,q, as the picture outsie of this strip consists only of vertical lines, an also label the horizontal number lines by the

20 20 JONATHAN BRUNDAN AND CATHARINA STROPPEL associate block sequence Γ = Γ 5 Γ 0. Γ 0 t 1 Γ 1 t 2 Γ 2 t 3 Γ 3 t 4 Γ 4 t 5 Γ 5 By a generalise cap in a stretche cap iagram we mean a component that meets the bottom number line at two ifferent vertices. An oriente stretche cap iagram is a consistently oriente iagram of the form t[γ] = γ t γ 1 γ 1 t 1 γ 0 where γ = γ γ 0 is a sequence of weights chosen from the associate block sequence Γ = Γ Γ 0, i.e. γ r Γ r for each r = 0,...,. In other wors, we ecorate the number lines of t by weights from the appropriate blocks, in such a way that the resulting iagram is consistently oriente. (For a precise efinition of the term oriente we refer to [BS1, 2]). Theorem 3.6. There are G-moule isomorphisms V(λ p,q ) V = P(λ) imp,q(λ), T p,q = λ Λ, ht(λ)=ht(λ p,q)+ λ Λ p,q, ht(λ)=ht(λ p,q)+ P(λ) imp,q(λ), where im p,q (λ) is the number of oriente stretche cap iagrams t[γ] of height such that γ 0 = λ p,q, γ = λ, an all generalise caps are anti-clockwise. Proof. For the first isomorphism, in view of Theorem 2.14 an Corollary 2.9, it suffices to prove the analogous statement on the iagram algebra sie, namely, that F i V (λ p,q ) = P (λ) imp,q(λ) (3.10) i Z λ Λ, ht(λ)=ht(λ p,q)+ as K-moules. Remembering that V (λ p,q ) = P (λ p,q ), this follows as an application of [BS2, Theorem 4.2], first using [BS2, Theorem 3.5] an [BS2, Theorem 3.6] to write the composite projective functor F i = F i F i1 in terms of inecomposable projective functors. The proof of the secon isomorphism is similar, taking only i (I p,q ) in (3.10). It is helpful to note that if λ Λ p,q an t[γ] is one of the oriente stretche cap iagrams counte by im p,q (λ) then t[γ] is trivial outsie the strip containing the vertices inexe by I p,q, + i.e. it consists only of straight lines oriente outsie that region. This follows by consiering (2.2). Corollary 3.7. The moules {P(λ) λ Λ p,q, ht(λ) = ht(λ p,q ) + } give a complete set of representatives for the isomorphism classes of inecomposable irect summans of T p,q.

21 KHOVANOV S DIAGRAM ALGEBRA IV 21 Proof. Suppose we are given λ Λ p,q with ht(λ) = ht(λ p,q ) +. Applying Corollary 2.13, there is a sequence i = (i 1,..., i ) (I p,q ) such that P(λ) is a summan of F i V(λ p,q ). Hence P(λ) is a summan of T p,q. Conversely, applying Theorem 3.6, we take λ Λ p,q with ht(λ) = ht(λ p,q ) + an im p,q (λ) 0, an must show that λ Λ p,q. There exists an oriente stretche cap iagram t[γ] of height with γ 0 = λ p,q an γ = λ, all of whose generalise caps are anti-clockwise. Every vertex labelle in λ must be at the left en of one of these anti-clockwise generalise caps, the right en of which gives a vertex labelle inexe by an integer q + n. Recalling the efinition (2.9), these observations prove that λ Λ p,q. Corollary 3.8. T p,q = {0} for > (m + n)(q p) + 2mn. Proof. The set Λ p,q has a unique element µ p,q of maximal height, namely, the weight p m }{{}}{{} n Using this an Theorem 3.6, we euce that T p,q = {0} for > ht(µ p,q ) ht(λ p,q ) = (m + n)(q p) + 2mn. The mirror image of the oriente stretche cap iagram u[δ] in a horizontal axis is enote u [δ ]. We call it an oriente stretche cup iagram. Then an oriente stretche circle iagram of height means a composite iagram of the form u [δ ] t[γ] = δ 0 u 1δ 1 δ 1 u γ t γ 1 γ 1 t 1 γ 0 where t[γ] an u[δ] are oriente stretche cap iagrams of height with γ = δ ; see [BS3, (6.17)] for an example. Theorem 3.9. The imension of the algebra En G (T p,q )op is equal to the number of oriente stretche circle iagrams u [δ ] t[γ] of height such that γ 0 = δ 0 = λ p,q an γ = δ Λ p,q. Proof. Applying Theorem 3.6, we see that the imension of the enomorphism algebra is equal to im p,q (λ) im p,q (µ) im Hom G (P(λ), P(µ)). λ,µ Λ p,q, ht(λ)=ht(µ)=ht(λ p,q)+ Also in view of Theorem 2.14, im Hom G (P(λ), P(µ)) = [P(µ) : L(λ)] is equal to the analogous imension im Hom K (P (λ), P (µ)) = [P (µ) : L(λ)] on the iagram algebra sie, which is escribe explicitly by [BS1, (5.9)]. We euce that im Hom G (P(λ), P(µ)) is equal to the number of weights ν such that λ ν µ an the circle iagram λνµ is consistently oriente. The theorem follows easily on combining this with the combinatorial efinitions of im p,q (λ) an im p,q (µ) from Theorem 3.6. The algebra R p,q an the isomorphism theorem. Now we nee to recall some of the main results of [BS3] which give an alternative iagrammatic escription of the algebra 1 p,q Hp,q. This will allow us to see to start with that m q+n