Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup
aa r X i v : . [ m a t h . R T ] N ov HIGHEST WEIGHT CATEGORIES ARISING FROMKHOVANOV’S DIAGRAM ALGEBRA IV: THE GENERALLINEAR SUPERGROUP
JONATHAN BRUNDAN AND CATHARINA STROPPEL
Abstract.
We prove that blocks of the general linear supergroup are Moritaequivalent to a limiting version of Khovanov’s diagram algebra. We deducethat blocks of the general linear supergroup are Koszul.
Contents
1. Introduction 12. Combinatorics of Grothendieck groups 83. Cyclotomic Hecke algebras and level two Schur-Weyl duality 164. Morita equivalence with generalised Khovanov algebras 305. Direct limits 35Index of notation 43References 431.
Introduction
This is the culmination of a series of four articles studying various general-isations of Khovanov’s diagram algebra from [Kh]. The goal is to relate thelimiting version H ∞ r of this algebra constructed in [BS1] to blocks of the gen-eral linear supergroup GL ( m | n ). More precisely, working always over a fixedalgebraically closed field F of characteristic zero, we show that any block of GL ( m | n ) of atypicality r is Morita equivalent to the algebra H ∞ r . We refer thereader to the introduction of [BS1] for a detailed account of our approach to thedefinition of Khovanov’s diagram algebra and the construction of its limitingversion; see also [St3] which discusses further the connections to link homology.To formulate our main result in detail, fix m, n ≥ G denote thealgebraic supergroup GL ( m | n ) over F . Using scheme-theoretic language, G canbe regarded 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 g = (cid:18) a bc d (cid:19) (1.1) Mathematics Subject Classification : 17B10, 16S37.First author supported in part by NSF grant no. DMS-0654147. where a (resp. d ) is an m × m (resp. n × n ) matrix with entries in A ¯0 , and b (resp. c ) is an m × n (resp. n × m ) matrix with entries in A ¯1 .We are interested here in finite dimensional representations of G , which canbe viewed equivalently as integrable supermodules over its Lie superalgebra g ∼ = gl ( m | n, F ); the condition for integrability is the same as for g ¯0 ∼ = gl ( m, F ) ⊕ gl ( n, F ). For example, we have the natural G -module V of column vectors, withstandard basis v , . . . , v m , v m +1 , . . . , v m + n and Z -grading defined by putting v r in degree ¯ r := ¯0 if 1 ≤ r ≤ m , ¯ r := ¯1 if m + 1 ≤ r ≤ m + n . Let B and T be thestandard choices of Borel subgroup and maximal torus: for each commutativesuperalgebra A , the groups B ( A ) and T ( A ) consist of all matrices g ∈ G ( A )that are upper triangular and diagonal, respectively. Let ε , . . . , ε m + n be theusual basis for the character group X ( T ) of T , i.e. ε r picks out the r th diagonalentry of a diagonal matrix. Equip X ( T ) with a symmetric bilinear form ( ., . )such that ( ε r , ε s ) = ( − ¯ r δ r,s , and set ρ := m X r =1 (1 − r ) ε r + n X s =1 ( m − s ) ε m + s . (1.2)Let X + ( T ) := (cid:26) λ ∈ X ( T ) (cid:12)(cid:12)(cid:12)(cid:12) ( λ + ρ, ε ) > · · · > ( λ + ρ, ε m ) , ( λ + ρ, ε m +1 ) < · · · < ( λ + ρ, ε m + n ) (cid:27) (1.3)denote the set of dominant weights .We allow only even morphisms between G -modules, so that the category of allfinite dimensional G -modules is obviously an abelian category. Any G -module M decomposes as M = M + ⊕ M − , where M + (resp. M − ) is the G -submoduleof M spanned by the degree ¯ λ (resp. the degree (¯ λ + ¯1)) component of the λ -weight space of M for all λ ∈ X ( T ); here ¯ λ := ( λ, ε m +1 + · · · + ε m + n ) (mod 2).It follows that the category of all finite dimensional G -modules decomposes as F ⊕ Π F , where F = F ( m | n ) (resp. Π F = Π F ( m | n )) is the full subcategoryconsisting of all M such that M = M + (resp. M = M − ). Moreover F and Π F are obviously equivalent. In view of this decomposition, we will focus just on F from now on. Note further that F is closed under tensor product, and itcontains both the natural module V and its dual V ∗ .By [B1, Theorem 4.47], the category F is a highest weight category withweight poset ( X + ( T ) , ≤ ), where ≤ is the Bruhat ordering defined combinatori-ally in the next paragraph. We fix representatives {L ( λ ) | λ ∈ X + ( T ) } for theisomorphism classes of irreducible modules in F so that L ( λ ) is an irreducibleobject in F generated by a one-dimensional B -submodule of weight λ . Wealso denote the standard and projective indecomposable modules in the highestweight category F by {V ( λ ) | λ ∈ X + ( T ) } and {P ( λ ) | λ ∈ X + ( T ) } , respectively.So P ( λ ) ։ V ( λ ) ։ L ( λ ). In this setting, the standard module V ( λ ) is oftenreferred to as a Kac module after [Ka].Now we turn our attention to the diagram algebra side. Let Λ = Λ( m | n )denote the set of all weights in the diagrammatic sense of [BS1, §
2] drawn ona number line with vertices indexed by Z , such that a total of m vertices arelabelled × or ∨ , a total of n vertices are labelled ◦ or ∨ , and all of the (infinitelymany) remaining vertices are labelled ∧ . From now on, we identify the set HOVANOV’S DIAGRAM ALGEBRA IV 3 X + ( T ) introduced above with the set Λ via the following weight dictionary .Given λ ∈ X + ( T ), we define I ∨ ( λ ) := { ( λ + ρ, ε ) , . . . , ( λ + ρ, ε m ) } , (1.4) I ∧ ( λ ) := Z \ { ( λ + ρ, ε m +1 ) , . . . , ( λ + ρ, ε m + n ) } . (1.5)Then we identify λ with the element of Λ whose i th vertex is labelled ◦ if i does 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 ) and I ∧ ( λ ). (1.6)For example, the zero weight (which parametrises the trivial G -module) is iden-tified with the diagram if m ≥ n ,if m ≤ n . · · · · · · n z }| { m − n z }| { ∧∧ ∨ ∨ ∨ × × × ∧ ∧ · · · · · · | {z } m | {z } n − m ∧∧ ∨ ∨ ∨ ◦ ◦ ◦ ∧ ∧ where the leftmost ∨ is on vertex (1 − m ). In these diagrammatic terms, theBruhat ordering on X + ( T ) mentioned earlier is the same as the Bruhat orderingon Λ from [BS1, § ≤ on diagrams generated by thebasic operation of swapping a ∨ and an ∧ so that ∨ ’s move to the right.Let ∼ be the equivalence relation on Λ generated by permuting ∨ ’s and ∧ ’s.Following the language of [BS1] again, the ∼ -equivalence classes of weights fromΛ are called blocks . The defect def(Γ) of each block Γ ∈ Λ / ∼ is simply equalto the number of vertices labelled ∨ in any weight λ ∈ Γ; this is the same thingas the usual notion of atypicality in the representation theory of GL ( m | n ) as ine.g. [Se1, (1.1)].Let K = K ( m | n ) denote the direct sum of the diagram algebras K Γ associ-ated to all the blocks Γ ∈ Λ / ∼ as defined in [BS1, § K has a basis { ( aλb ) | for all oriented circle diagrams aλb with λ ∈ Λ } , (1.7)and its multiplication is defined by an explicit combinatorial procedure in termsof such diagrams as in [BS1, § § λ ∈ Λ there is associated an idempotent e λ ∈ K . The leftideal P ( λ ) := Ke λ is a projective indecomposable module with irreducible headdenoted L ( λ ). The modules { L ( λ ) | λ ∈ Λ } are all one dimensional and give acomplete set of irreducible K -modules. Finally let V ( λ ) be the standard modulecorresponding to λ , which was referred to as a cell module in [BS1, § Theorem 1.1.
There is an equivalence of categories E from F ( m | n ) to thecategory of finite dimensional left K ( m | n ) -modules, such that E L ( λ ) ∼ = L ( λ ) , E V ( λ ) ∼ = V ( λ ) and E P ( λ ) ∼ = P ( λ ) for each λ ∈ Λ( m | n ) . JONATHAN BRUNDAN AND CATHARINA STROPPEL
Our proof of Theorem 1.1 involves showing that K is isomorphic to the locallyfinite endomorphism algebra End finG ( P ) op of a canonical minimal projectivegenerator P ∼ = L λ ∈ Λ P ( λ ) for F ; see Lemmas 5.8–5.9 below. To construct P ,we first consider the weight λ p,q := m X r =1 pε r − n X s =1 ( q + m ) ε m + s (1.8)for integers p ≤ q . This is represented diagrammatically by p qp − m q + n · · · · · · | {z } m | {z } n ∧∧ ∧∧∧× × × ◦ ◦ ◦ ∧ ∧ (1.9)where the rightmost × is on vertex p and the rightmost ◦ is on vertex ( q + n ).The G -module V ( λ p,q ) is projective, hence the “tensor space” V ( λ p,q ) ⊗ V ⊗ d is projective for any d ≥
0. Moreover, any P ( λ ) appears as a summand of V ( λ p,q ) ⊗ V ⊗ d for suitable p, q and d . The key step in our approach is to computethe endomorphism algebra of V ( λ p,q ) ⊗ V ⊗ d for d ≥
0. For d ≤ min( m, n ), weshow that it is a certain degenerate cyclotomic Hecke algebra of level two, givinga new “super” version of the level two Schur-Weyl duality from [BK1]. Thenwe invoke results from [BS3] which show that the basic algebra that is Moritaequivalent to this cyclotomic Hecke algebra is a generalised Khovanov algebra;this equivalence relies in particular on the connection between cyclotomic Heckealgebras and Khovanov-Lauda-Rouquier algebras in type A from [BK2]. Finallywe let p, q and d vary, taking a suitable direct limit to derive 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 allΓ ∈ Λ( m | n ) / ∼ are the blocks of the algebra K ( m | n ). Hence by Theorem 1.1they are the basic algebras representing the individual blocks of the category F ( m | n ). In the diagrammatic setting, it is obvious for Γ ∈ Λ( m | n ) / ∼ andΓ ′ ∈ Λ( m ′ | n ′ ) / ∼ (for possibly different m ′ and n ′ ) that the algebras K Γ and K Γ ′ are isomorphic if and only if Γ and Γ ′ have the same defect. Thus werecover a result of Serganova from [Se2]: the blocks of GL ( m | n ) for all m, n depend up to equivalence only on the degree of atypicality of the block. Gradings on blocks and Koszulity.
Each of the algebras K Γ carries a canon-ical positive grading 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 ) areKoszul. The appearence of such hidden Koszul gradings in representation the-ory goes back to the classic paper of Beilinson, Ginzburg and Soergel [BGS] onblocks of category O for a semisimple Lie algebra. In that work, the gradingis of geometric origin, whereas in our situation we establish the Koszulity in apurely algebraic way. Rigidity of Kac modules.
Another consequence of Theorem 1.1, combinedwith [BS2, Corollary 6.7] on the diagram algebra side, is that all the Kac mod-ules V ( λ ) are rigid , i.e. their radical and socle filtrations coincide. See [BS1,Theorem 5.2] for the explicit combinatorial description of the layers. HOVANOV’S DIAGRAM ALGEBRA IV 5
Kostant modules and BGG resolution.
In [BS2] we studied in detail the
Kostant modules for the generalised Khovanov algebras, i.e. the irreduciblemodules whose Kazhdan-Lusztig polynomials are multiplicity-free. In particu-lar in [BS2, Lemma 7.2] we classified the highest weights of these modules via apattern avoidance condition. Combining this with Theorem 1.1, we obtain thefollowing classification of all Kostant modules for GL ( m | n ): they are the irre-ducible modules parametrised by the weights in which no two vertices labelled ∨ have a vertex labelled ∧ between them. By [BS2, Theorem 7.3], Kostantmodules possess a BGG resolution by multiplicity-free direct sums of standardmodules. All irreducible polynomial representations of GL ( m | n ) satisfy thecombinatorial criterion to be Kostant modules, so this gives another proof ofthe main result of [CKL]. Endomorphism algebras of PIMs.
For any λ ∈ Λ, Theorem 1.1 impliesthat the endomorphism algebra End G ( P ( λ )) op of the projective indecomposablemodule P ( λ ) is isomorphic to the algebra e λ Ke λ . By the definition of multipli-cation in K , this algebra is isomorphic to F [ x , . . . , x r ] / ( x , . . . , x r ) where r isthe defect (atypicality) of the block containing λ , answering a question raisedrecently by several authors; see [BKN, (4.2)] and [Dr, Conjecture 4.3.3]. (Itshould also be possible to give a proof of the commutativity of these endomor-phism algebras using some deformation theory like in [St2, § P ( λ ) : V ( µ )) are at most one by Theorem 2.1 below;see [St1, Theorem 7.1] for a similar situation.) Super duality.
When combined with the results from [BS3], our results can beused to prove the “Super Duality Conjecture” as formulated in [CWZ]. A directalgebraic proof of this conjecture, and its substantial generalisation from [CW],has recently been found by Cheng and Lam [CL]. All of these results suggestsome more direct geometric connection between the representation theory of GL ( m | n ) and the category of perverse sheaves on Grassmannians may exist.To conclude this introduction, we recall in more detail the definition of thealgebra K following [BS1, § m = n = r and focus just onthe principal block of G = GL ( r | r ), which is the basic example of a block ofatypicality r . The dominant weights in this block are the weights λ i ,...,i r := r X s =1 ( i s + s − ε s − ε r +1 − s ) ∈ X + ( T )parametrised by sequences i > · · · > i r of integers. According to the weightdictionary (1.6), the diagram for λ i ,...,i r has label ∨ at the vertices indexed by i , . . . , i r , and label ∧ at all remaining vertices. The corresponding block ofthe algebra K is exactly the algebra denoted K ∞ r in the introduction of [BS1].Theorem 1.1 (or rather the more precise Lemmas 5.8–5.9 below) asserts in thissituation that K ∞ r ∼ = End finG M i > ··· >i r P ( λ i ,...,i r ) ! op . (1.10)Our explicit basis of K ∞ r is given by the oriented circle diagrams from (1.7).These are obtained by taking the diagram of some λ i ,...,i r , then gluing r cups JONATHAN BRUNDAN AND CATHARINA STROPPEL and infinitely many rays to the bottom and r caps and infinitely many rays tothe top of the diagram so that ◮ every vertex meets exactly one cup or ray below and exactly one cap orray above the number line; ◮ each cup and each cap is incident with one vertex labelled ∧ and onevertex labelled ∨ ; ◮ each ray is incident with a vertex labelled ∧ and extends from therevertically up or down to infinity; ◮ no crossings of cups, caps and rays are allowed.Under the isomorphism (1.10), such a diagram represents a homomorphism P ( λ j ,...,j r ) → P ( λ k ,...,k r ) where j > · · · > j r (resp. k > · · · > k r ) index theleftmost vertices of the cups (resp. caps) in the diagram. For example, here aretwo oriented circle diagrams corresponding to basis vectors in K ∞ (where · · · indicates infinitely many pairs of vertical rays labelled ∧ ): · · · · · · · · · · · · ∧ ∨ ∧ ∨ ∧ ∧ ✓ ✏✬ ✩✒ ✑✒ ✑ ∧ ∨ ∧ ∨ ∧ ∧ ✒ ✑✫ ✪✓ ✏✓ ✏ The first diagram here represents a homomorphism P ( λ , ) → P ( λ , ) and thesecond one represents a homomorphism P ( λ , ) → P ( λ , ). Multiplying thesetwo basis vectors together as described in the next paragraph, bearing in mindthe “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 , ) → P ( λ , ).We now sketch briefly the combinatorial procedure for multiplying basis vec-tors. Given two basis vectors, their product is necessarily zero unless the capsat the top of the first diagram are in exactly the same positions as the cups atthe bottom of the second. Assuming that is the case, we glue the first diagramunderneath the second and join matching pairs of rays. Then we perform asequence of generalised surgery procedures to smooth out all cup-cap pairs inthe symmetric middle section of the resulting composite diagram, obtainingzero or more new diagrams in which the middle section only involves verticalline segments. Finally we collapse these middle sections to obtain a sum ofbasis vectors, which is the desired product. Each generalised surgery procedurein this algorithm either involves two components in the diagram merging intoone or one component splitting into two. The rules for relabelling the newcomponent(s) produced when this operation is performed are summarized as HOVANOV’S DIAGRAM ALGEBRA IV 7 follows: 1 ⊗ , ⊗ x x, x ⊗ x , ⊗ y y, x ⊗ y , y ⊗ y , ⊗ x + x ⊗ , x x ⊗ x, y x ⊗ y, where 1 represents an anti-clockwise circle, x represents a clockwise circle, and y represents a line. This is a little cryptic; we refer the reader to [BS1] for a fulleraccount (and explanation of the connection to Khovanov’s original constructionvia a certain TQFT). Let us at least apply this algorithm to the example fromthe previous paragraph: two surgeries are needed, the first of which involves ananti-clockwise circle and a line merging together (1 ⊗ y y ) and the second ofwhich involves a line splitting into a clockwise circle and a line ( y x ⊗ y ): ∧ ∨ ∧ ∨ ∧ ∧ ✓✏✬ ✩✒✑✒✑ ∧ ∨ ∧ ∨ ∧ ∧ ✒✑✫ ✪✓✏✓✏ ∧ ∨ ∧ ∨ ∧ ∧ ✓✏✒✑✒✑ ∧ ∨ ∧ ∨ ∧ ∧ ✒✑✓✏✓✏ ∧ ∨ ∧ ∧ ∨ ∧ ✒✑✒✑ ∧ ∨ ∧ ∧ ∨ ∧ ✓✏✓✏ Contracting the middle section of the diagram on the right hand side here givesthe final product recorded already at the end of the previous paragraph.The case r = 1 in the above discussion (the principal block of GL (1 | { λ i | i ∈ Z } . It is wellknown that P ( λ i ) has irreducible socle and head isomorphic to L ( λ i ), withrad P ( λ i ) / soc P ( λ i ) ∼ = L ( λ i − ) ⊕ L ( λ i +1 ). Hence in this case the locally finiteendomorphism algebra from (1.10) has basis { e i , c i , a i , b i | i ∈ Z } , where e i isthe projection onto P ( λ i ), a i : P ( λ i ) → P ( λ i +1 ) and b i : P ( λ i +1 ) → P ( λ i ) arenon-zero homomorphisms chosen so that b i ◦ a i = a i − ◦ b i − , and c i := b i ◦ a i sends the head of P ( λ i ) onto its socle. This corresponds to our diagram basisfor K ∞ so that e i = ∧ ∧∨ ✒ ✑✓ ✏ c i = ∨∧ ∧ ✒ ✑✓ ✏ a i = ∨ ∧∧ ✓ ✏✒ ✑ b i = ∨ ∧∧ ✒ ✑✓ ✏ where we display only vertices i, i + 1 , i + 2 and there are infinitely many pairsof vertical rays labelled ∧ at all other vertices. In fact, K ∞ is simply the pathalgebra of the infinite quiver · · · • a i − ) ) • b i − l l a i ( ( • b i h h a i +1 ( ( • b i +1 h h a i +2 , , • · · · b i +2 i i modulo the relations a i b i = b i − a i − and a i a i +1 = 0 = b i +1 b i for all i ∈ Z . It isclear from the quiver description that K ∞ is naturally graded by path length;this is actually a Koszul grading. For general r the canonical Koszul gradingon K ∞ r is defined by declaring that a basis vector is of degree equal to the totalnumber of clockwise cups and caps in the oriented circle diagram. JONATHAN BRUNDAN AND CATHARINA STROPPEL
Acknowledgements.
This article was written up during stays by both authorsat the Isaac Newton Institute in Spring 2009. We thank the INI staff and theAlgebraic Lie Theory programme organisers for the opportunity.2.
Combinatorics of Grothendieck groups
In this preliminary section, we compare the combinatorics underlying therepresentation theory of GL ( m | n ) with that of the diagram algebra K ( m | n ).Our exposition is largely independent of [B1], indeed, we will reprove the rel-evant results from there as we go. On the other hand, we do assume thatthe reader is familiar with the general theory of diagram algebras developed in[BS1, BS2]. Later in the article we will also need to appeal to various resultsfrom [BS3]. Representation theory of K ( m | n ). Fix once and for all integers m, n ≥ K = K ( m | n ) and Λ = Λ( m | n ) be as in the introduction. The elements { e λ | λ ∈ Λ } form a system of (in general infinitely many) mutually orthogonalidempotents in K such that K = M λ,µ ∈ Λ e λ Ke µ . (2.1)So the algebra K is locally unital , but it is not unital (except in the trivial case m = n = 0). By a K -module we always mean a locally unital module; for a left K -module M this means that M decomposes as M = M λ ∈ Λ e λ M. The irreducible K -modules { L ( λ ) | λ ∈ Λ } defined in the introduction are allone dimensional, so K is a basic algebra.Let rep( K ) denote the category of finite dimensional left K -modules. TheGrothendieck group [rep( K )] of this category is the free Z -module on basis { [ L ( λ )] | λ ∈ Λ } . The standard modules { V ( λ ) | λ ∈ Λ } and the projectiveindecomposable modules { P ( λ ) | λ ∈ Λ } from [BS1, §
5] are finite dimensional,so it makes sense to consider their classes [ V ( λ )] and [ P ( λ )] in [rep( K )]. Finally,we use the notation µ ⊃ λ (resp. µ ⊂ λ ) from [BS1, §
2] to indicate that thecomposite diagram µλ (resp. µλ ) is oriented in the obvious sense. Theorem 2.1.
We have in [rep( K )] that [ P ( λ )] = X µ ⊃ λ [ V ( µ )] , [ V ( λ )] = X µ ⊂ λ [ L ( µ )] for each λ ∈ Λ .Proof. This follows from [BS1, Theorem 5.1] and [BS1, Theorem 5.2]. (cid:3) As µ ⊃ λ (resp. µ ⊂ λ ) implies that µ ≥ λ (resp. µ ≤ λ ) in the Bruhatordering, we deduce from Theorem 2.1 that the classes { [ P ( λ )] } and { [ V ( λ )] } are linearly independent in [rep( K )]. However they do not span [rep( K )] as thechains in the Bruhat order are infinite. HOVANOV’S DIAGRAM ALGEBRA IV 9
Remark 2.2.
The algebra K possesses a natural Z -grading defined by declaringthat each basis vector ( aλb ) from (1.7) is of degree equal to the number ofclockwise cups and caps in the diagram aλb . This means that one can considerthe graded representation theory of K . The various modules L ( λ ) , V ( λ ) and P ( λ ) also possess canonical gradings, as is discussed in detail in [BS1, § Special projective functors: the diagram side.
As in [BS3, (2.5)], let usrepresent a block Γ ∈ Λ / ∼ by means of its block diagram , that is, the diagramobtained by taking any λ ∈ Γ and replacing all the ∧ ’s and ∨ ’s labelling itsvertices by the symbol • . Because m and n are fixed, the block Γ can berecovered uniquely from its block diagram. Recall also the notion of the defect of a weight λ ∈ Λ from [BS1, § ∨ in λ , and the defect of λ is the same thing as the defectdef(Γ) of the unique block Γ ∈ Λ / ∼ containing λ .Given a block Γ, we say that i ∈ Z is Γ -admissible if the i th and ( i + 1)thvertices of the block diagram of Γ match the top number line of a unique oneof the following pictures, and def(Γ) is as indicated: Γ t i (Γ)Γ − α i ❄ F i def(Γ) ≥ ≥ ≥ ≥ “cup” “cap” “right-shift” “left-shift” ✒ ✑ × ◦ • • ✓ ✏ ◦ ו • ◦◦ ❅❅❅ •• ×× (cid:0)(cid:0)(cid:0) • • (2.2)Assuming i is Γ-admissible, we let (Γ − α i ) denote the block obtained from Γ byrelabelling the i th and ( i + 1)th vertices of its block diagram according to thebottom number line of the appropriate picture. Also define a (Γ − α i )Γ-matching t i (Γ) in the sense of [BS2, §
2] so that the strip between the i th and ( i + 1)thvertices of t i (Γ) is as in the picture, and there are only vertical “identity” linesegments elsewhere.For blocks Γ , ∆ ∈ Λ / ∼ and a Γ∆-matching t , recall the geometric bimodule K t Γ∆ from [BS2, § K Γ , K ∆ )-bimodule. We can viewit as a ( K, K )-bimodule by extending the actions of K Γ and K ∆ to all of K so that the other blocks act as zero. The functor K t Γ∆ ⊗ K ? is an endofunctorof rep( K ) called a projective functor . Writing t ∗ for the mirror image of t in ahorizontal axis, the functor K t ∗ ∆Γ ⊗ K ? gives another projective functor which isbiadjoint to K t Γ∆ ⊗ K ? by [BS2, Corollary 4.9].For any i ∈ Z , introduce the ( K, K )-bimodules e F i := M Γ K t i (Γ)(Γ − α i )Γ , e E i := M Γ K t i (Γ) ∗ Γ(Γ − α i ) , (2.3)where the direct sums are over all Γ ∈ Λ / ∼ such that i is Γ-admissible. The special projective functors are the endofunctors F i := e F i ⊗ K ? and E i := e E i ⊗ K ?of rep( K ) defined by tensoring with these bimodules. The discussion in theprevious paragraph implies that the functors F i and E i are biadjoint, hencethey are both exact and map projectives to projectives.For λ ∈ Λ, let I × ( λ ) := I ∨ ( λ ) ∩ I ∧ ( λ ) (resp. I ◦ ( λ ) := Z \ ( I ∨ ( λ ) ∪ I ∧ ( λ )))denote the set of integers indexing the vertices labelled × (resp. ◦ ) in λ ; cf. (1.6). Introduce the notion of the height of λ :ht( λ ) := X i ∈ I × ( λ ) i − X i ∈ I ◦ ( λ ) i. (2.4)Note all weights belonging to the same block have the same height. Lemma 2.3.
For λ ∈ Λ and 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 ) − ).Proof. This follows by inspecting (2.2). (cid:3)
Lemma 2.4.
Let λ ∈ Λ and i ∈ Z . For symbols x, y ∈ {◦ , ∧ , ∨ , ×} we write λ xy for the diagram obtained from λ by relabelling the i th and ( i + 1) th verticesby x and 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 → V ( λ ∧∨ ) → F i V ( λ ) → V ( λ ∨∧ ) → F i L ( λ ) has irreducible socle and head both isomorphic to L ( λ ∨∧ ) ,and 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 ◦ × ) and F i L ( λ ) ∼ = L ( λ ◦ × ) . (vii) If λ = λ ∧∨ then F i V ( λ ) ∼ = V ( λ ◦ × ) and F i L ( λ ) = { } . (viii) If λ = λ ∨∨ then F i V ( λ ) = F i L ( λ ) = { } . (ix) If λ = λ ∧∧ then F i V ( λ ) = F i L ( λ ) = { } . (x) For all other λ we have that F i P ( λ ) = F i V ( λ ) = F i L ( λ ) = { } .For the dual statement about E i , interchange all occurrences of ◦ and × .Proof. Apply [BS2, Theorems 4.2], [BS2, Theorem 4.5] and [BS2, Theorem4.11], exactly as was done in [BS3, Lemma 3.4]. (cid:3)
Remark 2.5.
Using Lemma 2.4, one can check that the endomorphisms of[rep( K )] induced by the functors F i and E i for all i ∈ Z satisfy the Serre re-lations defining the Lie algebra sl ∞ . Indeed, letting V ∞ denote the natural sl ∞ -module of column vectors, the category rep( K ) can be interpreted in aprecise sense as a categorification of a certain completion of the sl ∞ -module V m V ∞ ⊗ V n V ∗∞ ; see also [B1] where this point of view is taken on the super-group side. Using the graded representation theory mentioned in Remark 2.2,i.e. replacing rep( K ) with the category of finite dimensional graded K -modules,one gets a categorification of the q -analogue of this module over the quantisedenveloping algebra U q ( sl ∞ ); the action of q comes from shifting the gradingon a module up by one. We are not going to pursue this connection furtherhere, but refer the reader to [BS3, Theorem 3.5] where an analogous “gradedcategorification theorem” is discussed in detail. HOVANOV’S DIAGRAM ALGEBRA IV 11
The crystal graph.
Define the crystal graph to be the directed coloured graphwith vertex set equal to Λ and a directed edge 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 and only ifthe i th and ( i + 1)th vertices of λ and µ are labelled according to one of thesix cases in the following table, and all other vertices of λ and µ are labelled inthe same way: µ ∨ ◦ ∧ ◦ × ∨ × ∧ × ◦ ∨ ∧ λ ◦ ∨ ◦ ∧ ∨ × ∧ × ∨ ∧ ◦ × (2.5)Comparing this explicit description with [B1, § sl ∞ -modulementioned 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 − } , (2.6) I + p,q := { p − m + 1 , p − m + 2 , . . . , q + n − , q + n } . (2.7)(The reader may find it helpful at this point to note which vertices of the weight λ p,q from (1.9) are indexed by the set I + p,q .) Then introduce the following subsetsof Λ:Λ p,q := { λ ∈ Λ | the i th vertex of λ is labelled ∧ for all i / ∈ I + p,q } , (2.8)Λ ◦ p,q := n λ ∈ Λ p,q (cid:12)(cid:12)(cid:12) amongst vertices j, . . . , q + n of λ , the numberof ∧ ’s is ≥ the number of ∨ ’s, for all j ∈ I + p,q o . (2.9)Note that the weight λ p,q from (1.9) belongs to Λ ◦ p,q . It is the unique weight inΛ p,q of minimal height. Lemma 2.6.
Given λ ∈ Λ , choose p ≤ q such that λ ∈ Λ ◦ p,q (which is alwayspossible as there are infinitely many ∧ ’s and finitely many ∨ ’s). Then there areintegers i , . . . , i d ∈ I p,q , where d = ht( λ ) − ht( λ p,q ) , such that λ p,q i → · · · i d → λ is a path in the crystal graph. Moreover we have that F i d · · · F i V ( λ p,q ) ∼ = P ( λ ) ⊕ r , where r is the number of edges in the given path of the form ∨∧ → ◦ × .Proof. For the first statement, we proceed by induction on ht( λ ). If ht( λ ) =ht( λ p,q ), then λ = λ p,q and the conclusion is trivial. Now assume that ht( λ ) > ht( λ p,q ). As λ ∈ Λ ◦ p,q and λ = λ p,q , it is possible to find i ∈ I p,q such that the i thand ( i + 1)th vertices of λ are labelled ◦∨ , ◦∧ , ◦× , ∨× , ∧× 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 done by induction. To deduce the secondstatement, we apply Lemma 2.4 to get easily that F i d · · · F i P ( λ p,q ) ∼ = P ( λ ) ⊕ r .Finally P ( λ p,q ) ∼ = V ( λ p,q ) as λ p,q is of defect zero, by [BS1, Theorem 5.1]. (cid:3) Representation theory of GL ( m | n ). Now we turn to discussing the repre-sentation theory of G = GL ( m | n ). In the introduction, we defined already theabelian category F = F ( m | n ) and the irreducible modules {L ( λ ) } , the stan-dard modules {V ( λ ) } and the projective indecomposable modules {P ( λ ) } , all of which are parametrised by the set X + ( T ) of dominant weights. We are using anunusual font here (and a few other places later on) to avoid confusion with theanalogous K -modules { L ( λ ) } , { V ( λ ) } and { P ( λ ) } . Recall in particular thatthe Z -grading on L ( λ ) is defined so that its λ -weight space is concentrated indegree ¯ λ := ( λ, ε m +1 + · · · + ε m + n ) (mod 2). Bearing in mind that we consideronly even morphisms, the modules {L ( λ ) | λ ∈ X + ( T ) } ∪ { Π L ( λ ) | λ ∈ X + ( T ) } give a complete set of pairwise non-isomorphic irreducible G -modules, where Πdenotes the change of parity functor.The standard module V ( λ ) is usually called a Kac module in this settingafter [Ka], and can be constructed explicitly as follows. Let P be the parabolicsubgroup 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 ) denote the usual finite dimensional irreducible module of highest weight λ for the underlying even subgroup G ¯0 ∼ = GL ( m ) × GL ( n ), viewing E ( λ ) as asupermodule with Z -grading concentrated in degree ¯ λ . We can regard E ( λ )also as a P -module 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 and p denote the Lie superalgebras of G and P , respectively. Thisconstruction makes sense because the induced module on the right hand side of(2.10) is an integrable g -supermodule, i.e. it lifts in a unique way to a G -module.The module L ( λ ) is isomorphic to the unique irreducible quotient of V ( λ ).Also P ( λ ) is the projective cover of L ( λ ) in the category F . It has a standardflag , that is, a filtration whose sections are standard modules. The multiplicity( P ( λ ) : V ( µ )) of V ( µ ) as a section of any such standard flag is given by the BGG reciprocity formula ( P ( λ ) : V ( µ )) = [ V ( µ ) : L ( λ )] , (2.11)as follows from [Zo] or the discussion in [B2, Example 7.5]. Special projective functors: the supergroup side.
Recall the weightdictionary from (1.6) by means of which we identify the set X + ( T ) with theset Λ = Λ( m | n ). Under this identification, the usual notion of the degree ofatypicality of a weight λ ∈ X + ( T ) corresponds to the notion of defect of λ ∈ Λ.Given λ, µ ∈ Λ, the irreducible G -modules L ( λ ) and L ( µ ) have the same centralcharacter if and only if λ ∼ µ in the diagrammatic sense; this can be deducedfrom [Se1, Corollary 1.9]. Hence the category F decomposes as F = M Γ ∈ Λ / ∼ F Γ , (2.12)where F Γ is the full subcategory consisting of the modules all of whose compo-sition factors are of the form L ( λ ) for λ ∈ Γ. We let pr Γ : F → F be the exactfunctor defined by projection onto F Γ along (2.12).Recall that V denotes the natural G -module and V ∗ is its dual. Following[B1, (4.21)–(4.22)], we define the special projective functors F i and E i for each HOVANOV’S DIAGRAM ALGEBRA IV 13 i ∈ Z to be the following endofunctors of F : F i := M Γ pr Γ − α i ◦ (? ⊗ V ) ◦ pr Γ , E i := M Γ pr Γ ◦ (? ⊗ V ∗ ) ◦ pr Γ − α i , (2.13)where the direct sums are over all Γ ∈ Λ / ∼ such that i is Γ-admissible (as in(2.3)). The functors ? ⊗ V and ? ⊗ V ∗ are biadjoint, hence so are F i and E i . Inparticular, all these functors are exact and send projectives to projectives. Forlater use, let us fix once and for all a choice of an adjunction making ( F i , E i )into an adjoint pair for each i ∈ Z . Lemma 2.7.
The following hold 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 ) , arranged in order 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 ) , arranged in order from top to bottom.Proof. This follows from the definition (2.10) and the tensor identity. (cid:3)
Corollary 2.8.
The following hold for any λ ∈ X + ( T ) and 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 ) and ( λ + ρ, ε r ) = i + (1 − ( − ¯ r ) / , arrangedin order 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 ) and ( λ + ρ, ε r ) = i + (1 + ( − ¯ r ) / , arrangedin order from top to bottom.Proof. For (i), apply pr Γ − α i to the statement of Lemma 2.7(i), where Γ is theblock containing λ (and do a little work to translate the combinatorics). Theproof of (ii) is similar. (cid:3) Corollary 2.9.
We have that ? ⊗ V = L i ∈ Z F i and ? ⊗ V ∗ = L i ∈ Z E i . The next lemma gives an alternative definition of the functors F i and E i which will be needed in the next section; cf. [CW, Proposition 5.2]. LetΩ := m + n X r,s =1 ( − ¯ s e r,s ⊗ e s,r ∈ g ⊗ g , (2.14)where e r,s denotes the rs -matrix unit. This corresponds to the supertrace formon g , so left multiplication by Ω (interpreted with the usual superalgebra signconventions) defines a G -module endomorphism of M ⊗ N for any G -modules M and N . Lemma 2.10.
For any G -module M , we have that F i M (resp. E i M ) is thegeneralised i -eigenspace (resp. the generalised − ( m − n + i ) -eigenspace) of theoperator Ω acting on M ⊗ V (resp. M ⊗ V ∗ ).Proof. We just explain for F i . Let c := P m + nr,s =1 ( − ¯ s e r,s e s,r ∈ U ( g ) be theCasimir element. It acts on V ( λ ) by multiplication by the scalar c λ := ( λ + 2 ρ + ( m − n − δ, λ ) where δ = ε + · · · + ε m − ε m +1 − · · · − ε m + n . Also, we have that Ω =(∆( c ) − c ⊗ − ⊗ c ) / U ( g ). Now toprove the lemma, it suffices to verify it for the special case M = V ( λ ). Usingthe observations just made, we see that multiplication by Ω preserves the filtra-tion from Lemma 2.7(i), and the induced action of Ω on the section V ( λ + ε r )is by multiplication by the scalar( c λ + ε r − c λ − m + n ) / λ + ρ, ε r ) + (1 − ( − ¯ i ) / . The result follows on comparing with Corollary 2.8(i). (cid:3)
The next two lemmas are the key to understanding the representation theoryof GL ( m | n ) from a combinatorial point of view. Lemma 2.11.
Let i ∈ Z and λ ∈ Λ be a weight such that the i th and ( i + 1) thvertices of λ are labelled ∧ and ∨ , respectively. Let µ be the weight obtained from λ by interchanging the labels on these two vertices. Then L ( µ ) is a compositionfactor of V ( λ ) .Proof. This is a reformulation of [Se1, Theorem 5.5]. It can be proved directlyby an explicit calculation with certain lowering operators in U ( g ) as in [BS3,Lemma 4.8]. (cid:3) Lemma 2.12.
Exactly the same statement as Lemma 2.4 holds in the cate-gory F , replacing L ( λ ) , V ( λ ) , P ( λ ) , F i and E i by L ( λ ) , V ( λ ) , P ( λ ) , F i and E i ,respectively.Proof. The statements involving V ( λ ) follow from Corollary 2.8. The remainingparts then follow by mimicking the arguments used to prove [BS3, Lemma 4.9],using Lemma 2.11 in place of [BS3, Lemma 4.8]. (cid:3) Corollary 2.13.
Given λ ∈ Λ , pick p, q , d , i , . . . , i d and r as in Lemma 2.6.Then we have that F i d · · · F i V ( λ p,q ) ∼ = P ( λ ) L r .Proof. We note as λ p,q is of defect 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 corollaryfollows from Lemma 2.12 in exactly the same way that Lemma 2.6 was deducedfrom Lemma 2.4. (cid:3) Identification of Grothendieck groups.
Consider the Grothendieck group[ F ] of F . It is the free Z -module on basis { [ L ( λ )] | λ ∈ Λ } . The exact functors F i and E i (resp. F i and E i ) induce endomorphisms of the Grothendieck group[ F ] (resp. [rep( K )]), which we denote by the same notation. The last part ofthe following theorem recovers the main result of [B1]. Theorem 2.14.
Define a Z -module isomorphism ι : [ F ] ∼ → [rep( K )] by declar-ing that ι ([ L ( λ )]) = [ L ( λ )] for each λ ∈ Λ . (i) We have that ι ([ V ( λ )]) = [ V ( λ )] and ι ([ P ( λ )]) = [ P ( λ )] for each λ ∈ Λ . (ii) For each i ∈ Z , we have that F i ◦ ι = ι ◦ F i and E i ◦ ι = ι ◦ E i as linearmaps from [ F ] to [rep( K )] . HOVANOV’S DIAGRAM ALGEBRA IV 15 (iii)
We have in [ F ] that [ P ( λ )] = X µ ⊃ λ [ V ( µ )] , [ V ( λ )] = X µ ⊂ λ [ L ( µ )] for each λ ∈ Λ .Proof. Given λ ∈ Λ, let p, q , d , r and i , . . . , i d be as in Lemma 2.6. ByLemma 2.6 and Theorem 2.1, we know already that[ P ( λ )] = 12 r · F i d · · · F i [ V ( λ p,q )] = X µ ⊃ λ [ V ( µ )] , (2.15)all equalities written in [rep( K )]. In view of Lemma 2.12, the action of F i onthe classes of standard modules in [rep( K )] is described by exactly the samematrix as the action of F i on the classes of standard modules in [ F ]. So wededuce from the second equality in (2.15) that12 r · F i d · · · F i [ V ( λ p,q )] = X µ ⊃ λ [ V ( µ )] , equality in [ F ]. By Corollary 2.13 this also equals [ P ( λ )], proving the firstformula in (iii). The second formula in (iii) follows from the first and (2.11).Then (i) is immediate from the definition of ι and the coincidence of theformulae in (iii) and Theorem 2.1.Finally to deduce (ii), we have already noted that ι ( F i [ V ( λ )]) = F i [ V ( λ )] forevery λ . It follows easily from this that ι ( F i [ P ( λ )]) = F i [ P ( λ )] for every λ .Using also the adjointness of F i and E i (resp. F i and E i ) we deduce that[ E i L ( µ ) : L ( λ )] = dim Hom K ( P ( λ ) , E i L ( µ ))= dim Hom K ( F i P ( λ ) , L ( µ )) = dim Hom G ( F i P ( λ ) , L ( µ ))= dim 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 and E i . The argument for F i and F i is similar. (cid:3) Highest weight structure and duality.
At this point, we can also deducethe following result, which recovers [B1, Theorem 4.47].
Theorem 2.15.
The category F is a highest weight category in the sense of [CPS] with weight poset (Λ , ≤ ) . The modules {L ( λ ) } , {V ( λ ) } and {P ( λ ) } giveits irreducible, standard and projective indecomposable modules, respectively.Proof. We already noted just before (2.11) that P ( λ ) has a standard flag with V ( λ ) at the top. Moreover by Theorem 2.14(iii) all the other sections of thisflag are all of the form V ( µ ) with µ > λ in the Bruhat order. The theoremfollows from this, (2.11) and the definition of highest weight category. (cid:3) The costandard modules in the highest weight category F can be constructedexplicitly as the duals V ( λ ) ⊛ of the standard modules with respect to a natural duality ⊛ . This duality maps a G -module M to the linear dual M ∗ with theaction of G defined using the supertranspose anti-automorphism g g st , where g st = (cid:18) a t − c t b t d t (cid:19) for g of the form (1.1). Note ⊛ fixes irreducible modules, i.e. L ( λ ) ⊛ ∼ = L ( λ ) foreach λ ∈ Λ.3.
Cyclotomic Hecke algebras and level two Schur-Weyl duality
Fix integers p ≤ q and let λ p,q be the weight of defect zero from (1.8). Thestandard module V ( λ p,q ) is projective. As the functor ? ⊗ V sends projectivesto projectives, the G -module V ( λ p,q ) ⊗ V ⊗ d is again projective for any d ≥ Action of the degenerate affine Hecke algebra.
We begin by constructingan explicit basis for V ( λ p,q ) ⊗ V ⊗ d . Recalling (2.10), we have that V ( λ p,q ) = U ( g ) ⊗ U ( p ) E ( λ p,q ) . (3.1)Let det m (resp. det n ) denote the one-dimensional G ¯0 -module defined by tak-ing the determinant of GL ( m ) (resp. GL ( n )), with Z -grading concentrated indegree ¯0. Then the module E ( λ p,q ) in (3.1) is the inflation to P of the mod-ule Π n ( q + m ) (det pm ⊗ det − ( q + m ) n ), so it is also one dimensional. Hence, fixing anon-zero highest weight vector v p,q ∈ V ( λ p,q ), the induced module V ( λ p,q ) is ofdimension 2 mn with basis ( m + n Y r = m +1 m Y s =1 e τ r,s r,s · v p,q (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ r,s ≤ ) , (3.2)where the products here are taken in any fixed order (changing the order onlychanges the vectors by ± v , . . . , v m + n is the standard basisfor the natural module V , from which we get the obvious monomial basis { v i ⊗ · · · ⊗ v i d | ≤ i , . . . , i d ≤ m + n } (3.3)for V ⊗ d . Tensoring (3.2) and (3.3), we get the desired basis for V ( λ p,q ) ⊗ V ⊗ d .Now let H d be the degenerate affine Hecke algebra from [D]. This is theassociative algebra equal as a vector space to F [ x , . . . , x d ] ⊗ F S d , the tensorproduct of a polynomial algebra and the group algebra of the symmetric group S d . Multiplication is defined so that F [ x , . . . , x d ] ≡ F [ x , . . . , x d ] ⊗ F S d ≡ ⊗ F S d are subalgebras of H d , and 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 denotes the r th basic transposition ( r r + 1).By [CW, Proposition 5.1], there is a right action of H d on V ( λ p,q ) ⊗ V ⊗ d by G -module endomorphisms. The transposition s r acts as the “super” flip( v ⊗ v i ⊗· · ·⊗ v i r ⊗ v i r +1 ⊗· · ·⊗ v i d ) s r = ( − ¯ i r ¯ i r +1 v ⊗ v i ⊗· · ·⊗ v i r +1 ⊗ v i r ⊗· · ·⊗ v i d . This is the same as the endomorphism defined by left multiplication by theelement Ω from (2.14) so that the first and second tensors in Ω hit the ( r + 1)thand ( r + 2)th tensor positions in V ( λ p,q ) ⊗ V ⊗ d , respectively. The polynomial HOVANOV’S DIAGRAM ALGEBRA IV 17 generator x s acts by left multiplication by Ω so that the first tensor in Ω isspread across tensor positions 1 , . . . , s using the comultiplication of U ( g ) andthe second tensor in Ω hits the ( s + 1)th tensor position in V ( λ p,q ) ⊗ V ⊗ d . Thefollowing lemma gives an explicit formula for the action of x s in a special case. Lemma 3.1.
For ≤ i , . . . , i d ≤ m + n and ≤ s ≤ d , we have that ( v p,q ⊗ v i ⊗ · · · ⊗ v i d ) x s = pv p,q ⊗ v i ⊗ · · · ⊗ v i d + s − X r =1 ( − ¯ i r ¯ i s + P r The element ( x − p )( x − q ) ∈ H d acts as zero on V ( λ p,q ) ⊗ V ⊗ d .Proof. It suffices to check this in the special case that d = 1. In that case,Lemma 3.1 shows that( v p,q ⊗ v i ) x = pv p,q ⊗ v i if 1 ≤ i ≤ m , qv p,q ⊗ v i + P mj =1 ( − n ( q + m ) e i,j ( v p,q ⊗ v j ) if m + 1 ≤ i ≤ m + n .It follows easily that ( x − p )( x − q ) acts as zero on the vector v p,q ⊗ v i forevery 1 ≤ i ≤ m + n . These vectors generate V ( λ p,q ) ⊗ V as a G -module so wededuce that ( x − p )( x − q ) acts as zero the whole module. (cid:3) Corollary 3.3. If d ≤ min( m, n ) then the endomorphisms of V ( λ p,q ) ⊗ V ⊗ d defined by right multiplication by { x σ · · · x σ d d w | ≤ σ , . . . , σ d ≤ , w ∈ S d } arelinearly independent.Proof. Any vector v ∈ V ( λ p,q ) ⊗ V ⊗ d can be written as v = P i ∈ I b i ⊗ c i where { b i | i ∈ I } is the basis from (3.2) and the c i ’s are unique vectors in V ⊗ d . Werefer to c i as the b i -component of v . Exploiting the assumption on d , we can pick distinct integers m + 1 ≤ i , . . . , i d ≤ m + n and 1 ≤ j , . . . , j d ≤ m . Take0 ≤ σ , . . . , σ d ≤ v p,q ⊗ v i ⊗ · · · ⊗ v i d ) x σ · · · x σ d d . For 0 ≤ τ , . . . , τ d ≤ 1, Lemma 3.1 implies that the e τ i ,j · · · e τ d i d ,j d · v p,q -componentof ( v p,q ⊗ v i ⊗ · · · ⊗ v i d ) x σ · · · x σ d d is zero either if τ + · · · + τ d > σ + · · · + σ d , orif τ + · · · + τ d = σ + · · · + σ d but τ r = σ r for some r . Moreover, if τ r = σ r forall r , then the e τ i ,j · · · e τ d i d ,j d · v p,q -component of ( v p,q ⊗ v i ⊗ · · · ⊗ v i d ) x σ · · · x σ d d is equal to ± v k ⊗ · · · ⊗ v k d where k r = i r if σ r = 0 and k r = j r if σ r = 1.This is enough to show that the vectors ( v p,q ⊗ v i ⊗ · · · ⊗ v i d ) x σ · · · x σ d d w forall 0 ≤ σ , . . . , σ d ≤ w ∈ S d are linearly independent, and the corollaryfollows. (cid:3) In view of Corollary 3.2, the right action of H d on V ( λ p,q ) ⊗ V ⊗ d induces anaction of the quotient algebra H p,qd := H d / h ( x − p )( x − q ) i . (3.4)This algebra is a particular example of a degenerate cyclotomic Hecke algebra of level two. It is well known (e.g. see [BK1, Lemma 3.5]) that dim H p,qd = 2 d d !. Corollary 3.4. If d ≤ min( m, n ) the action of H p,qd on V ( λ p,q ) ⊗ V ⊗ d is faithful.Proof. This follows on comparing the dimension of H p,qd with the number oflinearly independent endomorphisms constructed in Corollary 3.3. (cid:3) Since the action of H p,qd on V ( λ p,q ) ⊗ V ⊗ d is by G -module endomorphisms,it induces an algebra homomorphismΦ : H p,qd → End G ( V ( λ p,q ) ⊗ V ⊗ d ) op . (3.5)The main goal in the remainder of the section is to show that this homorphismis surjective . Weight idempotents and the space T p,qd . For a tuple i = ( i , . . . , i d ) ∈ Z d ,there is an idempotent e ( i ) ∈ H p,qd determined uniquely by the property thatmultiplication by e ( i ) projects any H p,qd -module onto its i -weight space , that is,the simultaneous generalised eigenspace for the commuting operators x , . . . , x d and eigenvalues i , . . . , i d , respectively. All but finitely many of the e ( i )’s arezero, and the non-zero ones give a system of mutually orthogonal idempotentsin H p,qd summing to 1; see e.g. [BK2, § e ( i ) on the module V ( λ p,q ) ⊗ V ⊗ d can beinterpreted as follows. In view of Corollary 2.9, we have that V ( λ p,q ) ⊗ V ⊗ d = M i ∈ Z d F i V ( λ p,q ) (3.6)where F i denotes the composite F i d ◦ · · · ◦ F i of the functors from (2.13).By Lemma 2.10 and the definition of the actions of x , . . . , x d , the summand F i V ( λ p,q ) in this decomposition is precisely the i -weight space of V ( λ p,q ) ⊗ V ⊗ d .Hence the weight idempotent e ( i ) acts on V ( λ p,q ) ⊗ V ⊗ d as the projection ontothe summand F i V ( λ p,q ) along the decomposition (3.6). HOVANOV’S DIAGRAM ALGEBRA IV 19 Recalling the interval I p,q from (2.6), we are usually from now on going torestrict our attention to the summand T p,qd := M i ∈ ( I p,q ) d F i V ( λ p,q ) (3.7)of V ( λ p,q ) ⊗ V ⊗ d . By the discussion in the previous paragraph, we have equiv-alently that T p,qd = ( V ( λ p,q ) ⊗ V ⊗ d )1 p,qd where1 p,qd := X i ∈ ( I p,q ) d e ( i ) ∈ H p,qd . (3.8)As a consequence of the fact that any symmetric polynomial in x , . . . , x d iscentral in H d , the idempotent 1 p,qd is central in H p,qd . The space T p,qd is naturallya right module over 1 p,qd H p,qd , which is a sum of blocks of H p,qd . Hence the mapΦ from (3.5) induces an algebra homomorphism1 p,qd H p,qd → End G ( T p,qd ) op . (3.9)As a refinement of the surjectivity of Φ proved below, we will also see later inthe section that the induced map (3.9) is an isomorphism . Note from (3.16)onwards we will denote the algebra 1 p,qd H p,qd instead by R p,qd . Stretched diagrams. In this subsection, we develop some combinatorial toolswhich will be used initially to compute the dimension of the various endomor-phism algebras that we are interested in. We say that a tuple i ∈ Z d is ( p, q ) -admissible if i r is Γ r − -admissible for each r = 1 , . . . , d , where Γ , . . . , Γ d aredefined recursively from Γ := { λ p,q } and Γ r := Γ r − − α i r , notation as in(2.2). We refer to the sequence Γ := Γ d · · · Γ Γ of blocks here as the associ-ated block sequence . The composite matching t = t d · · · t defined by setting t r := t i r (Γ r − ) for each r is the associated composite matching . Both of thesethings make sense only if i ∈ Z d is ( p, q )-admissible. Lemma 3.5. If i ∈ Z d is not ( p, q ) -admissible then F i V ( λ p,q ) is zero.Proof. This follows from the definitions and (2.13). (cid:3) By a stretched cap diagram t = t d · · · t of height d , we mean the associ-ated composite matching for some ( p, q )-admissible sequence i ∈ Z d . We canuniquely recover the sequence i , hence also the associated block sequence Γ ,from the stretched cap diagram t . Here is an example of a stretched cap di-agram of height 5, taking m = 2 , n = 1 and q − p = 1; we draw only thestrip containing the vertices indexed by I + p,q , as the picture outside of this stripconsists only of vertical lines, and also label the horizontal number lines by the associated block sequence Γ = Γ · · · Γ . Γ t Γ t Γ t Γ t Γ t Γ × × • ◦ ❅❅ × × • ◦ ✓✏ • •× • ✒✑ × ◦ × • (cid:0)(cid:0) × • ✓✏ ו• •• • By a generalised cap in a stretched cap diagram we mean a component thatmeets the bottom number line at two different vertices. An oriented stretchedcap diagram is a consistently oriented diagram of the form t [ γ ] = γ d t d γ d − · · · γ t γ where γ = γ d · · · γ is a sequence of weights chosen from the associated blocksequence Γ = Γ d · · · Γ , i.e. γ r ∈ Γ r for each r = 0 , . . . , d . In other words,we decorate the number lines of t by weights from the appropriate blocks, insuch a way that the resulting diagram is consistently oriented. (For a precisedefinition of the term oriented we refer to [BS1, § Theorem 3.6. There are G -module isomorphisms V ( λ p,q ) ⊗ V ⊗ d ∼ = M λ ∈ Λ , ht( λ )=ht( λ p,q )+ d P ( λ ) ⊕ dim p,q ( λ ) ,T p,qd ∼ = M λ ∈ Λ p,q , ht( λ )=ht( λ p,q )+ d P ( λ ) ⊕ dim p,q ( λ ) , where dim p,q ( λ ) is the number of oriented stretched cap diagrams t [ γ ] of height d such that γ = λ p,q , γ d = λ , and all generalised caps are anti-clockwise.Proof. For the first isomorphism, in view of Theorem 2.14 and Corollary 2.9, itsuffices to prove the analogous statement on the diagram algebra side, namely,that M i ∈ Z d F i V ( λ p,q ) ∼ = M λ ∈ Λ , ht( λ )=ht( λ p,q )+ d P ( λ ) ⊕ dim p,q ( λ ) (3.10)as K -modules. Remembering that V ( λ p,q ) = P ( λ p,q ), this follows as an appli-cation of [BS2, Theorem 4.2], first using [BS2, Theorem 3.5] and [BS2, Theo-rem 3.6] to write the composite projective functor F i = F i d ◦ · · · ◦ F i in termsof indecomposable projective functors.The proof of the second isomorphism is similar, taking only i ∈ ( I p,q ) d in(3.10). It is helpful to note that if λ ∈ Λ p,q and t [ γ ] is one of the orientedstretched cap diagrams counted by dim p,q ( λ ) then t [ γ ] is trivial outside thestrip containing the vertices indexed by I + p,q , i.e. it consists only of straightlines oriented ∧ outside that region. This follows by considering (2.2). (cid:3) Corollary 3.7. The modules {P ( λ ) | λ ∈ Λ ◦ p,q , ht( λ ) = ht( λ p,q ) + d } give acomplete set of representatives for the isomorphism classes of indecomposabledirect summands of T p,qd . HOVANOV’S DIAGRAM ALGEBRA IV 21 Proof. Suppose we are given λ ∈ Λ ◦ p,q with ht( λ ) = ht( λ p,q ) + d . ApplyingCorollary 2.13, there is a sequence i = ( i , . . . , i d ) ∈ ( I p,q ) d such that P ( λ ) is asummand of F i V ( λ p,q ). Hence P ( λ ) is a summand of T p,qd . Conversely, applyingTheorem 3.6, we take λ ∈ Λ p,q with ht( λ ) = ht( λ p,q ) + d and dim p,q ( λ ) = 0,and must show that λ ∈ Λ ◦ p,q . There exists an oriented stretched cap diagram t [ γ ] of height d with γ = λ p,q and γ d = λ , all of whose generalised caps areanti-clockwise. Every vertex labelled ∨ in λ must be at the left end of oneof these anti-clockwise generalised caps, the right end of which gives a vertexlabelled ∧ indexed by an integer ≤ q + n . Recalling the definition (2.9), theseobservations prove that λ ∈ Λ ◦ p,q . (cid:3) Corollary 3.8. T p,qd = { } for d > ( m + n )( q − p ) + 2 mn .Proof. The set Λ p,q has a unique element µ p,q of maximal height, namely, theweight p − m q + n · · · · · · | {z } n | {z } m ∧∧ ∧∧∧ ◦ ◦ ◦ × × × ∧ ∧ Using this and Theorem 3.6, we deduce that T p,qd = { } for d > ht( µ p,q ) − ht( λ p,q ) = ( m + n )( q − p ) + 2 mn . (cid:3) The mirror image of the oriented stretched cap diagram u [ δ ] in a horizontalaxis is denoted u ∗ [ δ ∗ ]. We call it an oriented stretched cup diagram . Then an oriented stretched circle diagram of height d means a composite diagram of theform u ∗ [ δ ∗ ] ≀ t [ γ ] = δ u ∗ δ · · · δ d − u ∗ d γ d t d γ d − · · · γ t γ where t [ γ ] and u [ δ ] are oriented stretched cap diagrams of height d with γ d = δ d ;see [BS3, (6.17)] for an example. Theorem 3.9. The dimension of the algebra End G ( T p,qd ) op is equal to thenumber of oriented stretched circle diagrams u ∗ [ δ ∗ ] ≀ t [ γ ] of height d such that γ = δ = λ p,q and γ d = δ d ∈ Λ p,q .Proof. Applying Theorem 3.6, we see that the dimension of the endomorphismalgebra is equal to X λ,µ ∈ Λ p,q , ht( λ )=ht( µ )=ht( λ p,q )+ d dim p,q ( λ ) · dim p,q ( µ ) · dim Hom G ( P ( λ ) , P ( µ )) . Also in view of Theorem 2.14, dim Hom G ( P ( λ ) , P ( µ )) = [ P ( µ ) : L ( λ )] is equalto the analogous dimension dim Hom K ( P ( λ ) , P ( µ )) = [ P ( µ ) : L ( λ )] on thediagram algebra side, which is described explicitly by [BS1, (5.9)]. We deducethat dim Hom G ( P ( λ ) , P ( µ )) is equal to the number of weights ν such that λ ∼ ν ∼ µ and the circle diagram λνµ is consistently oriented. The theoremfollows easily on combining this with the combinatorial definitions of dim p,q ( λ )and dim p,q ( µ ) from Theorem 3.6. (cid:3) The algebra R p,qd and the isomorphism theorem. Now we need to recallsome of the main results of [BS3] which give an alternative diagrammatic de-scription of the algebra 1 p,qd H p,qd . This will allow us to see to start with that this algebra has the same dimension as the endomorphism algebra from Theo-rem 3.9. For the reader wishing to understand already the relationship betweenthe diagram algebra R p,qd defined in the next paragraph and the algebra K ( m | n )from the introduction, we point to Lemma 4.1, (4.7) and Corollary 4.6 in thenext section. On the other hand Corollary 3.22 established later in this sectionexplains the connection between R p,qd and the representation theory of G .Let R p,qd be the associative, unital algebra with basis (cid:26) | u ∗ [ δ ∗ ] ≀ t [ γ ] | (cid:12)(cid:12)(cid:12)(cid:12) for all oriented stretched circle diagrams u ∗ [ δ ∗ ] ≀ t [ γ ]of height d with γ = δ = λ p,q and γ d = δ d ∈ Λ p,q (cid:27) . The multiplication is defined by an explicit algorithm described in detail in[BS3]. Briefly, to multiply two basis vectors | s ∗ [ τ ∗ ] ≀ r [ σ ] | and | u ∗ [ δ ∗ ] ≀ t [ γ ] | , theproduct is zero unless r = u and all mirror image pairs of internal circles in r [ σ ] and u ∗ [ δ ∗ ] are oriented so that one is clockwise, the other anti-clockwise.Assuming these conditions hold, the product is computed by putting s ∗ [ τ ∗ ] ≀ r [ σ ]underneath u ∗ [ δ ∗ ] ≀ t [ γ ], erasing all internal circles and number lines in r [ σ ]and u ∗ [ δ ∗ ], then iterating the generalised surgery procedure to smooth out thesymmetric middle section of the diagram. Lemma 3.10. The algebras R p,qd and End G ( T p,qd ) op have the same dimension.In particular, R p,qd is the zero algebra for d > ( m + n )( q − p ) + 2 mn .Proof. The number of elements in the diagram basis for R p,qd is the same asthe dimension of the algebra End G ( T p,qd ) op thanks to Theorem 3.9. The laststatement follows from Corollary 3.8. (cid:3) As a consequence of [BS3, Corollary 8.6], we can identify R p,qd with a certain cyclotomic Khovanov-Lauda-Rouquier algebra in the sense of [KL, Ro]. To makethis identification explicit, we need to define some special elements { e ( i ) | i ∈ ( I p,q ) d } ∪ { y , . . . , y d } ∪ { ψ , . . . , ψ d − } (3.11)in R p,qd . For i ∈ ( I p,q ) d , we let e ( i ) ∈ R p,qd be the idempotent defined as follows.If i is not ( p, q )-admissible then e ( i ) := 0. If it is admissible, let t = t d · · · t bethe associated composite matching and Γ = Γ d · · · Γ be the associated blocksequence. Then e ( i ) := X δ , γ | t ∗ [ δ ∗ ] ≀ t [ γ ] | (3.12)where the sum is over all sequences γ = γ d · · · γ and δ = δ d · · · δ of weightswith each γ r , δ r ∈ Γ r chosen so that every circle of t ∗ [ δ ∗ ] ≀ t [ γ ] crossing the middlenumber line is anti-clockwise, and all remaining mirror image pairs of circlesare oriented so that one is clockwise, the other anti-clockwise. The elements { e ( i ) | i ∈ ( I p,q ) d } give a system of mutually orthogonal idempotents whose sumis the identity in R p,qd .Next we define the elements y , . . . , y d . Let ¯ y , . . . , ¯ y d be the unique elementsof R p,qd such that the product | u ∗ [ δ ∗ ] ≀ t [ γ ] |· ¯ y r (resp. ¯ y r ·| u ∗ [ δ ∗ ] ≀ t [ γ ] | ) is computedby making a positive circle move in the section of u ∗ t containing t r (resp. u ∗ r ),as described in detail in [BS3, (5.5), (5.11)]. Also introduce the signs σ rp,q ( i ) := ( − min( p,i r )+min( q,i r )+ m − p − δ i ,ir −···− δ ir − ,ir . (3.13) HOVANOV’S DIAGRAM ALGEBRA IV 23 Then we define y r := P i ∈ ( I p,q ) d y r e ( i ) where y r e ( i ) := σ rp,q ( i )¯ y r e ( i ) , (3.14)to get the elements y r ∈ R p,qd for r = 1 , . . . , d .Finally we define ψ , . . . , ψ d − . Let ¯ ψ , . . . , ¯ ψ d − be the unique elementsof R p,qd such that the product | u ∗ [ δ ∗ ] ≀ t [ γ ] | · ¯ ψ r (resp. ¯ ψ r · | u ∗ [ δ ∗ ] ≀ t [ γ ] | ) iscomputed by making a negative circle move, a crossing move or a height movein the section of u ∗ t containing t r +1 t r (resp. u ∗ r u ∗ r +1 ), as described in detail in[BS3, (5.7),(5.12)]. Then we define ψ r := P i ∈ ( I p,q ) d ψ r e ( i ) where ψ r e ( i ) := (cid:26) − σ rp,q ( i ) ¯ ψ r e ( i ) if i r +1 = i r or i r +1 = i r + 1,¯ ψ r e ( i ) otherwise, (3.15)to get the elements ψ r ∈ R p,qd for r = 1 , . . . , d − Theorem 3.11. The elements (3.11) generate R p,qd subject only to the defin-ing relations of the Khovanov-Lauda-Rouquier algebra associated to the linearquiver • • • −→−→ · · · • • −→ with vertices indexed by the set I p,q in order from left to right (see e.g. [BS3,(6.8)–(6.16)] ), plus the additional cyclotomic relations y δ i ,p + δ i ,q e ( i ) = 0 for i = ( i , . . . , i d ) ∈ ( I p,q ) d .Proof. This is a consequence of [BS3, Corollary 8.6]. More precisely, we apply[BS3, Corollary 8.6], taking the index set I there to be the set I p,q , the pair( o + m, o + n ) there to be ( p, q ), and summing over all α ∈ Q + of height d .This implies that the given quotient of the Khovanov-Lauda-Rouquier algebrais isomorphic to the diagram algebra with basis consisting of oriented stretchedcircle diagrams | u ∗ [ δ ∗ ] ≀ t [ γ ] | just like the ones considered here, except they aredrawn only in the strip containing the vertices indexed by I + p,q . The isomorphismin [BS3] is not quite the same as the map here, because the sign in (3.13) differsfrom the corresponding sign chosen in [BS3] by a factor of ( − m − p ; this causesno problems as it amounts to twisting by an automorphism of the Khovanov-Lauda-Rouquier algebra. It remains to observe that all the oriented stretchedcircle diagrams in the statement of the present theorem are trivial outside thestrip I + p,q , consisting only of straight lines oriented ∧ in that region; these haveno effect on the multiplication. (cid:3) Now we can formulate the following key isomorphism theorem , which identi-fies the algebras 1 p,qd H p,qd and R p,qd . Theorem 3.12. There is a unique algebra isomorphism p,qd H p,qd ∼ → R p,qd such that e ( i ) e ( i ) , x r e ( i ) ( y r + i r ) e ( i ) and s r e ( i ) ( ψ r q r ( i ) − p r ( i )) e ( i ) for each r and i ∈ ( I p,q ) d , where p r ( i ) , q r ( i ) ∈ R p,qd are chosen as in [BK2, § , e.g. one could take p r ( i ) := (cid:26) if i r = i r +1 , − ( i r +1 − i r + y r +1 − y r ) − if i r = i r +1 ; q r ( i ) := y r +1 − y r if i r = i r +1 , (2 + y r +1 − y r )(1 + y r +1 − y r ) − if i r +1 = i r + 1 , if i r +1 = i r − , i r +1 − i r + y r +1 − y r ) − if | i r − i r +1 | > .(The inverses on the right hand sides of these formulae make sense because each y r +1 − y r is nilpotent with nilpotency degree at most two, as is clear from thediagrammatic definition of the y r ’s.)Proof. This is a consequence of Theorem 3.11 combined with the main theoremof [BK2]; see also [BS3, Theorem 8.5]. (cid:3) Henceforth, we will use the isomorphism from the above theorem to identify the algebra 1 p,qd H p,qd with R p,qd , so1 p,qd H p,qd ≡ R p,qd . (3.16)We will denote it always by the more compact notation R p,qd . Thus there arethree different ways of viewing R p,qd : it is a diagram algebra with basis given byoriented stretched circle diagrams, it is a cyclotomic Khovanov-Lauda-Rouquieralgebra, and it is a sum of blocks of the cyclotomic Hecke algebra H p,qd . Super version of level two Schur-Weyl duality. Now we can prove themain results of the section, namely, that the map Φ from (3.5) is surjective andthe induced map (3.9) is an isomorphism. In the case d ≤ min( m, n ) we havealready done most of the work: Theorem 3.13. If d ≤ min( m, n ) then we have that T p,qd = V ( λ p,q ) ⊗ V ⊗ d , R p,qd = H p,qd , and the map Φ : H p,qd → End G ( V ( λ p,q ) ⊗ V ⊗ d ) op is an algebra isomorphism.Proof. Let us first show that T p,qd = V ( λ p,q ) ⊗ V ⊗ d . Observe for d ≤ min( m, n )that any ( p, q )-admissible sequence i ∈ Z d necessarily lies in ( I p,q ) d . This isclear from (2.2) and the form of the diagram λ p,q . Hence applying Lemma 3.5we get that F i V ( λ p,q ) = { } for i ∈ Z d \ ( I p,q ) d . So we are done by (3.7).Now consider the map Φ. It is injective by Corollary 3.4. To show that itis an isomorphism, we apply Lemma 3.10, recalling the identification (3.16), tosee thatdim End G ( V ( λ p,q ) ⊗ V ⊗ d ) op = dim End G ( T p,qd ) op = dim R p,qd ≤ dim H p,qd . Hence our injective map is an isomorphism. At the same time, we deduce thatdim R p,qd = dim H p,qd , hence R p,qd = H p,qd . (cid:3) It remains to consider the cases with d > min( m, n ). For that, following astandard argument, we need to allow m and n to vary. So we take some otherintegers m ′ , n ′ ≥ G ′ = GL ( m ′ | n ′ ). To avoidany confusion, we decorate all notation related to G ′ with a prime, e.g. V ′ is HOVANOV’S DIAGRAM ALGEBRA IV 25 its natural module, its irreducible modules are the modules denoted L ′ ( λ ) for λ ∈ Λ ′ := Λ( m ′ | n ′ ), and F ′ := F ( m ′ | n ′ ). We are going to exploit the followingstandard lemma. Lemma 3.14. Let F : F ′ → F be an exact functor and X ⊆ Λ ′ be a subsetwith the following properties: (i) F commutes with duality, i.e. F ◦ ⊛ ∼ = ⊛ ◦ F ; (ii) the modules F V ′ ( λ ) for λ ∈ X have standard flags; (iii) the map Hom G ′ ( V ′ ( λ ) , V ′ ( µ ) ⊛ ) → Hom G ( F V ′ ( λ ) , F V ′ ( µ ) ⊛ ) defined bythe functor F is surjective for all λ, µ ∈ X .Suppose M, N are G ′ -modules with standard flags all of whose sections are ofthe form V ′ ( λ ) for λ ∈ X . Then the map Hom G ′ ( M, N ⊛ ) → Hom G ( F M, F N ⊛ ) defined by the functor F is surjective.Proof. We proceed by induction on the sum of the lengths of the standard flagsof M and N , the base case being covered by (iii). For the induction step, either M or N has a standard flag of length greater than one. It suffices to considerthe case when the standard flag of M has length greater than 1, since the othercase reduces to that using duality. Pick a submodule K of M such that both K and Q := M/K are non-zero and possess standard flags. By a general factabout highest weight categories (see also [B2, Lemma 3.6] for a short directproof in this context), the functor Hom G ′ (? , N ⊛ ) is exact on sequences of G ′ -modules possessing a standard flag. So applying it to the short exact sequence0 → K → M → Q → → Hom G ′ ( Q, N ⊛ ) → Hom G ′ ( M, N ⊛ ) → Hom G ′ ( K, N ⊛ ) → y y y → Hom G ( F Q, F N ⊛ ) → Hom G ( F M, F N ⊛ ) → Hom G ( F K, F N ⊛ ) → . Similar considerations applying the functor Hom G (? , F N ⊛ ) to 0 → F K → F M → F Q → F . Theleft and right vertical arrows are surjective by the induction hypothesis. Hencethe middle vertical arrow is surjective too by the five lemma. (cid:3) Now we consider the situation that G ′ = GL ( m | n + 1) . (3.17)Embed G = GL ( m | n ) into G ′ in the top left hand corner in the obvious way.Also let S be the one-dimensional torus embedded into G ′ in the bottom righthand corner, so that S centralises the subgroup G . The character group X ( S )is generated by ε m + n +1 . Let F + : F ′ → F be the functor mapping M ∈ F ′ tothe − ( q + m ) ε m + n +1 -weight space of Π q + m M with respect to the torus S . Lemma 3.15. Let G ′ = GL ( m | n + 1) and G = GL ( m | n ) as in (3.17). (i) For λ ∈ X + ( T ′ ) with ( λ, ε m + n +1 ) = q + m , we have that F + V ′ ( λ ) ∼ = V ( µ ) , where µ is the restriction of λ to T < T ′ . (ii) F + V ′ ( λ ′ p,q ) ∼ = V ( λ p,q ) . (iii) For λ ∈ X + ( T ) with ( λ, ε m + n +1 ) < q + m , we have that F + V ′ ( λ ) = { } .Proof. The proof of (i) reduces easily using the definition (2.10) to checking thatthe − ( q + m ) ε m + n +1 -weight space of Π q + m E ′ ( λ ) with respect to S is isomorphicto E ( µ ) as a G ¯0 -module, which is well known. Then (ii) is a consequence of (i),noting that λ ′ p,q = P mr =1 pε r − P n +1 s =1 ( q + m ) ε m + s . The proof of (iii) is similarto (i). (cid:3) Lemma 3.16. Let G ′ = GL ( m | n + 1) and G = GL ( m | n ) as in (3.17). Thereis a unique G -module isomorphism F + ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) ∼ → V ( λ p,q ) ⊗ V ⊗ d such that v ′ p,q ⊗ v ′ i ⊗· · ·⊗ v ′ i d v p,q ⊗ v i ⊗· · ·⊗ v i d for all ≤ i , . . . , i d ≤ m + n .Moreover, this map intertwines the natural actions of H p,qd .Proof. The first statement follows using the isomorphism F + V ′ ( λ ′ p,q ) ∼ = V ( λ p,q )from the first part of the previous lemma, together with the following observa-tions: ◮ the weights µ arising with non-zero multiplicity in V ′ ( λ ′ p,q ) all satisfy( µ, ε m + n +1 ) ≤ q + m ; ◮ the weights µ arising with non-zero multiplicity in ( V ′ ) ⊗ d all satisfy( µ, ε m + n +1 ) ≤ ◮ the zero weight space of ( V ′ ) ⊗ d with respect to S is V ⊗ d .The second statement is straightforward. (cid:3) Lemma 3.17. Let G ′ = GL ( m | n + 1) and G = GL ( m | n ) as in (3.17). Assumethe map Φ ′ : H p,qd → End G ′ ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) op is surjective. Then the map Φ : H p,qd → End G ( V ( λ p,q ) ⊗ V ⊗ d ) op is surjective too.Proof. We apply Lemma 3.14 to F := F + , taking m ′ := m, n ′ := n + 1 and theset of weights X to be { λ ∈ X + ( T ′ ) | ( λ, ε m + n +1 ) ≤ q + m } . The hypothesisin Lemma 3.14(ii) follows from Lemma 3.15, and the other two hypotheses areclear. Since V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d is self-dual and has a standard flag, we deducethat the functor F + defines a surjectionEnd G ′ ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) op ։ End G ( F + ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d )) op . Composing with the isomorphism from Lemma 3.16 and using also the last partof that lemma, we deduce that there is a commutative triangle H p,qd Φ ′ ւ ց Φ End G ′ ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) op −−−−→ End G ( V ( λ p,q ) ⊗ V ⊗ d ) op in which the bottom map is surjective. The map Φ ′ is surjective by assumption.So we deduce that Φ is surjective too. (cid:3) Instead consider the situation that G ′ = GL ( m + 1 | n ) (3.18)and embed G = GL ( m | n ) into G ′ into the bottom right hand corner in theobvious way. Also let S be the one-dimensional torus embedded into G ′ in the HOVANOV’S DIAGRAM ALGEBRA IV 27 top left hand corner, so again S centralises the subgroup G . The charactergroup X ( S ) is generated by ε . Let F − : F ′ → F be the functor mapping M ∈ F ′ to the ( p − n ) ε -weight space of M with respect to the torus S . Theanalogues of Lemmas 3.15–3.17 in the new situation are as follows. Lemma 3.18. Let G ′ = GL ( m + 1 | n ) and G = GL ( m | n ) as in (3.18). (i) For λ ∈ X + ( T ′ ) with ( λ, ε ) = p , we have that F − V ′ ( λ ) ∼ = V ( µ ) , where µ is the restriction of λ − nε + ε m +2 + · · · + ε m + n +1 to T . (ii) F − V ′ ( λ ′ p,q ) ∼ = V ( λ p,q ) . (iii) For λ ∈ X + ( T ′ ) , with ( λ, ε ) > p , we have that F − V ′ ( λ ) = { } .Proof. This follows by similar arguments to the proof of Lemma 3.15, but thereis an additional subtlety. The main new point is that if v + is a non-zero highestweight vector of weight λ in V ′ ( λ ) as in (i), then the vector e ,m +2 · · · e ,m + n +1 v + gives a highest weight vector for G in F − V ′ ( λ ) of weight λ − nε + ε m +2 + · · · + ε m + n +1 . This statement is checked by explicit calculation in U ( g ′ ). Itthen follows from the PBW theorem that this vector generates F − V ′ ( λ ) and F − V ′ ( λ ) ∼ = V ( µ ) to give (i). For (ii) we note that λ ′ p,q = P m +1 r =1 pε r − P ns =1 ( q + m + 1) ε m +1+ s . (cid:3) Lemma 3.19. Let G ′ = GL ( m + 1 | n ) and G = GL ( m | n ) as in (3.18). Thereis a unique G -module isomorphism F − ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) ∼ → V ( λ p,q ) ⊗ V ⊗ d such that e ,m +2 · · · e ,m + n +1 · v ′ p,q ⊗ v ′ i ⊗ · · · ⊗ v ′ i d v p,q ⊗ v i − ⊗ · · · ⊗ v i d − for all ≤ i , . . . , i d ≤ m + n + 1 . Moreover, this map intertwines the naturalactions of H p,qd .Proof. Similar to the proof of Lemma 3.16, using Lemma 3.18. (cid:3) Lemma 3.20. Let G ′ = GL ( m + 1 | n ) and G = GL ( m | n ) as in (3.18). Assumethe map Φ ′ : H p,qd → End G ′ ( V ′ ( λ ′ p,q ) ⊗ ( V ′ ) ⊗ d ) op is surjective. Then the map Φ : H p,qd → End G ( V ( λ p,q ) ⊗ V ⊗ d ) op is surjective too.Proof. Apply Lemma 3.14 taking m ′ := m + 1 , n ′ := n and the set X of weightsto be { λ ∈ X + ( T ′ ) | ( λ, ε ) ≥ p } , arguing in the same way as in the proof ofLemma 3.17. (cid:3) Finally we can assemble the pieces to prove a key result, which is a superanalogue of the Schur-Weyl duality for level two from [BK1]. Theorem 3.21 (Super Schur-Weyl duality) . For any d ≥ , the map Φ : H p,qd → End G ( V ( λ p,q ) ⊗ V ⊗ d ) op is surjective.Proof. In the case that d ≤ min( m, n ), this is immediate from Theorem 3.13.To prove it in general, pick m ≤ m ′ and n ≤ n ′ so that d ≤ min( m ′ , n ′ ). Wealready know the surjectivity of the map Φ ′ for G ′ = GL ( m ′ | n ′ ). Now applyLemma 3.17 a total of ( n ′ − n ) times and Lemma 3.20 a total of ( m ′ − m ) timesto deduce the surjectivity for G = GL ( m | n ). (cid:3) Corollary 3.22. Recalling the identification (3.16), the map Φ induces analgebra isomorphism R p,qd ∼ → End G ( T p,qd ) op .Proof. Theorem 3.21 shows the induced map R p,qd → End G ( T p,qd ) op is surjective.It is an isomorphism by Lemma 3.10. (cid:3) Irreducible representations of R p,qd . As an application of Corollary 3.22,we can recover the known classification of the irreducible R p,qd -modules. For λ ∈ Λ ◦ p,q with ht( λ ) = ht( λ p,q ) + d , let D p,q ( λ ) := Hom G ( T p,qd , L ( λ )) , (3.19)viewed as a left R p,qd -module in the natural way. Theorem 3.23. The modules { D p,q ( λ ) | λ ∈ Λ ◦ p,q , ht( λ ) = ht( λ p,q ) + d } givea complete set of pairwise inequivalent irreducible R p,qd -modules. Moreover,we have that dim D p,q ( λ ) = dim p,q ( λ ) , where dim p,q ( λ ) is as defined in The-orem 3.6.Proof. As T p,qd is a projective module, Corollary 3.22 and the usual theory offunctors of the form Hom( P, ?) imply that the non-zero modules of the formHom G ( T p,qd , L ( λ )) for λ ∈ Λ give a complete set of pairwise inequivalent ir-reducible R p,qd -modules. The non-zero ones are parametrised by the weights λ ∈ Λ ◦ p,q with ht( λ ) = ht( λ p,q ) + d , thanks to Corollary 3.7. Finally the stateddimension formula is a consequence of Theorem 3.6. (cid:3) Remark 3.24. For a graded version of the dimension formula for the irreducible R p,qd -modules derived in Theorem 3.23, we refer the reader to [BS3, Theorem9.9]. (The identification of the labellings of irreducible representations in theabove theorem with the one in [BS3] can be deduced using the methods of thenext subsection.) i -Restriction and i -induction. To identify the labelling of irreducible R p,qd -modules from Theorem 3.23 with other known parametrisations, it is usefulto have available a more intrinsic characterisation of D p,q ( λ ). We explain oneinductive approach to this here in terms of the well-known i -restriction functors .Suppose that i ∈ I p,q . The natural inclusion H d ֒ → H d +1 induces anembedding H p,qd ֒ → H p,qd +1 . Composing before and after with the inclusion R p,qd = 1 p,qd H p,qd ֒ → H p,qd +1 and the projection H p,qd +1 ։ p,qd +1 H p,qd +1 = R p,qd +1 , weget a unital algebra homomorphism θ d : R p,qd → R p,qd +1 . (3.20)Note this map need not be injective, e.g. if d = ( m + n )( p − q ) + 2 mn then thealgebra R p,qd is non-zero but R p,qd +1 is zero by Lemma 3.10. The image of x d +1 in R p,qd +1 centralises θ d ( R p,qd ). So it makes sense to define the i -restriction functor E i : rep( R p,qd +1 ) → rep( R p,qd ) (3.21)to be the exact functor mapping an R p,qd +1 -module M to the generalised i -eigenspace of x d +1 on M , viewed as an R p,qd -module via θ d . HOVANOV’S DIAGRAM ALGEBRA IV 29 For us, a slightly different formulation of this definition will be more conve-nient. Let 1 p,qd ; i := X i ∈ ( I p,q ) d +1 , i d +1 = i e ( i ) ∈ R p,qd +1 . (3.22)Multiplication by this idempotent projects any R p,qd +1 -module M onto the gen-eralised i -eigenspace of x d +1 , which is a module over the subring 1 p,qd ; i R p,qd +1 p,qd ; i of R p,qd +1 . As 1 p,qd ; i centralises the image of the homomorphism θ d , we can defineanother unital algebra homomorphism θ d ; i : R p,qd → p,qd ; i R p,qd +1 p,qd ; i , x θ d ( x )1 p,qd ; i . (3.23)Because 1 p,qd +1 = P i ∈ I p,q p,qd ; i , we have that θ d = P i ∈ I p,q θ d ; i . The functor E i from the previous paragraph can be defined equivalently as the functor map-ping an R p,qd +1 -module M to the space 1 p,qd ; i M viewed as an R p,qd -module via thehomomorphism θ d ; i . So: E i M = 1 p,qd ; i M ∼ = Hom R p,qd +1 ( R p,qd +1 p,qd ; i , M ) , (3.24)where we view R p,qd +1 p,qd ; i as an ( R p,qd +1 , R p,qd )-bimodule using the homomorphism θ d ; i to get the right module structure. It is clear from (3.24) that the i -restrictionfunctor E i has a left adjoint F i := R p,qd +1 p,qd ; i ⊗ R p,qd ? : rep( R p,qd ) → rep( R p,qd +1 ) . (3.25)We refer to this as the i -induction functor . Lemma 3.25. There is an isomorphism r : E i ◦ Hom G ( T p,qd +1 , ?) ∼ → Hom G ( T p,qd , ?) ◦ E i of functors from F to rep( R p,qd ) .Proof. Take M ∈ F . Note recalling (3.6) that T p,qd +1 p,qd ; i = F i T p,qd . So we canidentify E i (Hom G ( T p,qd +1 , M )) = 1 p,qd ; i Hom G ( T p,qd +1 , M )= Hom G ( T p,qd +1 p,qd ; i , M ) = Hom G ( F i T p,qd , M ) . Then the adjunction between F i and E i fixed earlier defines a natural isomor-phism Hom G ( F i T p,qd , M ) ∼ → Hom G ( T p,qd , E i M ). Naturality gives automaticallythat this is an R p,qd -module homorphism. So we have defined the desired iso-morphism of functors r . (cid:3) Corollary 3.26. Take λ ∈ Λ ◦ p,q with ht( λ ) = ht( λ p,q ) + d + 1 for some d ≥ .ick µ ∈ Λ ◦ p,q such that µ i → λ is an edge in the crystal graph for some i ∈ I p,q .Then D p,q ( λ ) is the unique irreducible representation of R p,qd +1 with the propertythat E i D p,q ( λ ) has a quotient isomorphic to D p,q ( µ ) .Proof. By Lemma 2.12, we know for λ as in the statement of the corollary and i ∈ I p,q that E i L ( λ ) is zero unless there exists µ ∈ Λ ◦ p,q with µ i → λ in thecrystal graph, in which case L ( µ ) is the unique irreducible quotient of E i L ( λ ).The corollary follows from this on applying the exact functor Hom G ( T p,qd , ?)and using Lemma 3.25 and the definition (3.19). (cid:3) Remark 3.27. It is sometimes necessary to understand the homomorphism θ d ; i from (3.23) from a diagrammatic point of view. Using Theorem 3.12, wecan easily write down θ d ; i on the Khovanov-Lauda-Rouquier generators: it isthe map θ d ; i : e ( i ) e ( i + i ) , y r e ( i ) y r e ( i + i ) , ψ s e ( i ) ψ s e ( i + i ) (3.26)for i ∈ ( I p,q ) d , 1 ≤ r ≤ d and 1 ≤ s < d , where i + i denotes the ( d + 1)-tuple( i , . . . , i d , i ). It is harder to see θ d ; i in terms of the bases of oriented stretchedcircle diagrams, but this is worked out in detail in [BS3, Corollary 6.12]. Thebasic idea to compute θ d ; i ( | u ∗ [ γ ∗ ] ≀ t [ δ ] | ) is to insert two extra levels chosen from ✒ ✑ • • ✓ ✏ ◦ ו • ✒ ✑✓ ✏ × ◦ × ◦ ∨ ∧ ◦ ❅❅❅ • ◦◦ (cid:0)(cid:0)(cid:0) • • ×× ❅❅❅ ••× (cid:0)(cid:0)(cid:0) • into the middle of the matching u ∗ t , where we display only the strip betweenthe i th and ( i + 1)th vertices, the diagrams being trivial outside that strip.In the first configuration here, this process involves making one application ofthe generalised surgery procedure. The construction is made precise in theparagraph after [BS3, (6.34)].4. Morita equivalence with generalised Khovanov algebras Next we construct an explicit Morita equivalence between R p,q := L d ≥ R p,qd and a certain generalised Khovanov algebra K p,q . Using this, we replace thetensor space T p,q := L d ≥ T p,qd from the level two Schur-Weyl duality witha new space P p,q whose endomorphism algebra is K p,q . Exploiting the factthat K p,q is a basic algebra, we show that the space P p,q has exactly the sameindecomposable summands as T p,q (up to isomorphism), but that they eachappear with multiplicity one. Generalised Khovanov algebras. Given p ≤ q , let K p,q denote the subring e p,q Ke p,q of K , where e p,q is the (non-central) idempotent e p,q := X λ ∈ Λ ◦ p,q e λ ∈ K. (4.1) Lemma 4.1. The algebra K is the union of the subalgebras K p,q for all p ≤ q .Proof. This follows from (2.1) and the observation that, for any λ, µ ∈ Λ, wecan find integers p ≤ q such that both λ and µ belong to Λ ◦ p,q . (cid:3) Remark 4.2. In terms of the diagram basis from (1.7), K p,q has basis ( ( aλb ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for all oriented circle diagrams aλb with λ ∈ Λ p,q such thatcups and caps pass only through vertices in the interval I + p,q ) . All the diagrams in the diagram basis of K p,q consist simply of straight linesoriented ∧ outside of the interval I + p,q ; these play no role when computing themultiplication. So we can just ignore all of the diagram outside this stripwithout changing the algebra structure. This shows that the algebra K p,q is a HOVANOV’S DIAGRAM ALGEBRA IV 31 direct sum of the generalised Khovanov algebras from [BS1, § 6] associated tothe weights obtained from Λ p,q by erasing vertices outside the interval I + p,q . Representations of K p,q . To understand the representation theory of thealgebra K p,q , we exploit the exact functor e p,q : rep( K ) → rep( K p,q ) (4.2)arising by left multiplication by the idempotent e p,q ; cf. [BS1, (6.13)]. Itis easy to see that e p,q L ( λ ) = { } if and only if λ ∈ Λ ◦ p,q . Hence, letting L p,q ( λ ) := e p,q L ( λ ) for λ ∈ Λ ◦ p,q , the modules { L p,q ( λ ) | λ ∈ Λ ◦ p,q } (4.3)give a complete set of pairwise inequivalent irreducible K p,q -modules.Recalling also the ( K, K )-bimodules e F i and e E i from (2.3), we get ( K p,q , K p,q )-bimodules e F p,qi := e p,q e F i e p,q , e E p,qi := e p,q e E i e p,q (4.4)for any i ∈ I p,q . Let F i := e F p,qi ⊗ K p,q ? and E i := e E p,qi ⊗ K p,q ? be the endofunctorsof rep( K p,q ) defined by tensoring with these bimodules. Lemma 4.3. For any i ∈ I p,q , there are isomorphisms F i ◦ e p,q ∼ = e p,q ◦ F i and E i ◦ e p,q ∼ = e p,q ◦ E i of functors from rep( K ) to rep( K p,q ) .Proof. We just explain the proof for E i , since the argument for F i is the same.Suppose first that P is any projective right K -module that is isomorphic to adirect sum of summands of e p,q K . Then the natural multiplication map P e p,q ⊗ e p,q Ke p,q e p,q K → P is an isomorphism of right K -modules. This follows because it is obviously trueif P = e p,q K . In the next paragraph, we show that P = e p,q e E i satisfies thehypothesis that it is isomorphic to a direct sum of summands of e p,q K as aright K -module. Hence, we deduce that the multiplication map e E p,qi ⊗ K p,q e p,q K ∼ → e p,q e E i (4.5)is a ( K p,q , K )-bimodule isomorphism. The desired isomorphism E i ◦ e p,q ∼ = e p,q ◦ E i follows at once, since E i ◦ e p,q is the functor defined by tensoringwith the bimodule on the left hand side and e p,q ◦ E i is the functor defined bytensoring with the bimodule on the right hand side of (4.5).It remains to show that e p,q e E i is isomorphic to a direct sum of summandsof e p,q K . Equivalently, twisting with the obvious anti-automorphism ∗ thatreflects diagrams in a horizontal axis, we show that e F i e p,q is isomorphic to adirect sum of summands of Ke p,q . The indecomposable summands of Ke p,q areall of the form P ( µ ) for µ ∈ Λ ◦ p,q , so using the definition (4.1) this follows ifwe can show for any λ ∈ Λ ◦ p,q that all indecomposable summands of e F i e λ are ofthe form P ( µ ) for µ ∈ Λ ◦ p,q . As e F i e λ ∼ = F i P ( λ ), this follows easily from [BS2,Theorem 4.2], using also the assumption that i ∈ I p,q . (cid:3) Corollary 4.4. Let λ, µ and i be as in the statement of Corollary 3.26. Then L p,q ( λ ) is the unique irreducible representation of K p,q with the property that E i L p,q ( λ ) has a quotient isomorphic to L p,q ( µ ) . Proof. This follows from Lemmas 2.4 and 4.3 by the same argument used toprove Corollary 3.26. (cid:3) Morita bimodules. Recall the G -module T p,qd from (3.7). In view of Theorem3.21, we can identify its endomorphism algebra with the algebra R p,qd . Actuallyit is convenient now to work with all d simultaneously, setting T p,q := M d ≥ T p,qd , (4.6) R p,q := M d ≥ R p,qd ≡ End G ( T p,q ) op . (4.7)Note by Corollary 3.8 and Lemma 3.10 that T p,q and R p,q are both finite di-mensional.We next want to explain how the algebra R p,q is Morita equivalent to thebasic algebra K p,q , by writing down an explicit pair of bimodules A p,q and B p,q that induce the Morita equivalence. To do this, recall the notions oforiented upper- and lower-stretched circle diagrams from [BS3, (6.17)]. Theyare the consistently oriented diagrams obtained by gluing a cup diagram belowan oriented stretched cap diagram, or gluing a cap diagram above an orientedstretched cup diagram, respectively. Let A p,q and B p,q be the vector spaceswith bases (cid:26) ( a t [ γ ] | (cid:12)(cid:12)(cid:12)(cid:12) for all oriented upper-stretched circle diagrams a t [ γ ] ofheight d ≥ γ = λ p,q and γ d ∈ Λ p,q (cid:27) , (cid:26) | u ∗ [ δ ∗ ] b ) (cid:12)(cid:12)(cid:12)(cid:12) for all oriented lower-stretched circle diagrams u ∗ [ δ ∗ ] b ofheight d ≥ δ = λ p,q and δ d ∈ Λ p,q (cid:27) , respectively. We make A p,q into a ( K p,q , R p,q )-bimodule as follows. ◮ The left action of a basis vector ( aλb ) ∈ K p,q on ( c t [ γ ] | ∈ A p,q is byzero unless λ ∼ γ d (where γ = γ d · · · γ ) and b = c ∗ . Assuming theseconditions hold, the product is computed by drawing aλb underneath c t [ γ ], then iterating the generalised surgery procedure to smooth outthe symmetric middle section of the diagram. ◮ The right action of a basis vector | s ∗ [ τ ∗ ] ≀ r [ σ ] | ∈ R p,q on ( a t [ γ ] | ∈ A p,q is by zero unless t = s and all mirror image pairs of internal circlesin s ∗ [ τ ∗ ] and t [ γ ] are oriented so that one is clockwise, the other anti-clockwise. Assuming these conditions hold, the product is computed bydrawing a t [ γ ] underneath s ∗ [ τ ∗ ] ≀ r [ σ ], erasing all internal circles andnumber lines in t [ γ ] and s ∗ [ τ ∗ ], then iterating the generalised surgeryprocedure in the middle section once again.Similarly we make B p,q into an ( R p,q , K p,q )-bimodule. We refer the readerto [BS3, § 6] for detailed proofs (in an entirely analogous setting) that thesebimodules are well defined. Theorem 4.5. There are isomorphisms µ : A p,q ⊗ R p,q B p,q ∼ → K p,q , ν : B p,q ⊗ K p,q A p,q ∼ → R p,q of ( K p,q , K p,q ) -bimodules and of ( R p,q , R p,q ) -bimodules, respectively. HOVANOV’S DIAGRAM ALGEBRA IV 33 Proof. This is a consequence of [BS3, Theorem 6.2] and [BS3, Remark 6.7].These references give a somewhat indirect construction of the desired isomor-phisms µ and ν . The same maps can also be constructed much more directlyby mimicking the definitions of multiplication in the algebras R p,q and K p,q ,respectively. (cid:3) Corollary 4.6 (Morita equivalence) . The bimodule B p,q is a projective genera-tor for rep( R p,q ) . Also there is an algebra isomorphism K p,q ∼ → End R p,q ( B p,q ) op induced by the right action of K p,q on B p,q . Hence the functors Hom R p,q ( B p,q , ?) : rep( R p,q ) → rep( K p,q ) ,B p,q ⊗ K p,q ? : rep( K p,q ) → rep( R p,q ) are quasi-inverse equivalences of categories.Proof. This follows immediately from the theorem by the usual arguments ofthe Morita theory; see e.g. [Ba, (3.5) Theorem]. (cid:3) More about i -restriction and i -induction. We will view the i -restrictionand i -induction functors E i and F i from (3.24)–(3.25) now as endofunctors ofrep( R p,q ). Summing the maps θ d ; i from (3.23) over all d ≥ 0, we get a unitalalgebra homomorphism θ i : R p,q → p,qi R p,q p,qi where 1 p,qi := X d ≥ p,qd ; i (4.8)(which makes sense as the sum has only finitely many non-zero terms). Then E i is the functor defined by multiplying by the idempotent 1 p,qi , viewing the resultas an R p,q -module via θ i . The i -induction functor F i = R p,q p,qi ⊗ R p,q ? is leftadjoint to E i ; here, we are viewing R p,q p,qi as a right R p,q -module via θ i . Lemma 4.7. There is an isomorphism s ′ : B p,q ⊗ K p,q ? ◦ E i ∼ → E i ◦ B p,q ⊗ K p,q ? of functors from rep( K p,q ) to rep( R p,q ) .Proof. By the definitions of the various functors, it suffices to construct an( R p,q , K p,q )-bimodule isomorphism B p,q ⊗ K p,q e E p,qi ∼ → p,qi B p,q , where 1 p,qi B p,q is viewed as a left R p,q -module via the homomorphism θ i . Thereis an obvious multiplication map defined on a tensor product of basis vectors ofthe form | u ∗ [ δ ∗ ] b ) ⊗ ( cλtµd ) so that it is zero unless c = b ∗ and λ ∼ δ d (where δ = δ d · · · δ ), in which case it is the sum of basis vectors obtained by applyingthe generalised surgery procedure to the bc -part of the diagram obtained byputting u ∗ [ δ ∗ ] b underneath cλtµd . The fact that this multiplication map isan isomorphism of right K p,q -modules is a consequence of [BS2, Theorem 3.5].It remains to show that it is a left R p,q -module homomorphism. Using thediagrammatic description of the map θ i from Remark 3.27, this reduces tochecking a statement which, on applying the anti-automorphism ∗ , is equivalentto the identity (6.38) established in the proof of [BS3, Theorem 6.11]. (cid:3) Corollary 4.8. There is an isomorphism s : E i ◦ Hom R p,q ( B p,q , ?) ∼ → Hom R p,q ( B p,q , ?) ◦ E i of functors from rep( R p,q ) to rep( K p,q ) .Proof. In view of Corollary 4.6, the natural transformations arising from thecanonical adjunction between tensor and hom give isomorphisms of functors η : Id rep( K p,q ) ∼ → Hom R p,q ( B p,q , ?) ◦ B p,q ⊗ K p,q ? ,ε : B p,q ⊗ K p,q ? ◦ Hom R p.q ( B p,q , ?) ∼ → Id rep( R p,q ) . Now take the isomorphism from Lemma 4.7, compose on the left and the rightwith the functor Hom R p,q ( B p,q , ?), then use the isomorphisms η and ε to cancelthe resulting pairs of quasi-inverse functors. (cid:3) Identification of irreducible representations. Now we can identify thelabelling of the irreducible R p,q -modules from Lemma 3.23 with the labelling ofthe irreducible K p,q -modules from (4.3). Lemma 4.9. For λ ∈ Λ ◦ p,q , we have that Hom R p,q ( B p,q , D p,q ( λ )) ∼ = L p,q ( λ ) as K p,q -modules.Proof. We first show that L := Hom R p,q ( B p,q , D p,q ( λ p,q )) ∼ = L p,q ( λ p,q ). It isobvious that E i D p,q ( λ p,q ) = { } for all i ∈ I p,q . So by Corollary 4.8 we getthat E i L = { } for all i ∈ I p,q . Combined with Corollary 4.4, this implies that L ∼ = L p,q ( µ ) for some µ ∈ Λ ◦ p,q with ht( µ ) = ht( λ p,q ), hence L ∼ = L p,q ( λ p,q ) as λ p,q is the only such weight µ .Now take λ ∈ Λ ◦ p,q different from λ p,q , so that ht( λ ) > ht( λ p,q ). We againneed to show that L := Hom R p,q ( B p,q , D p,q ( λ )) ∼ = L p,q ( λ ). Let µ and i beas in Corollary 3.26, so D p,q ( µ ) is a quotient of E i D p,q ( λ ). We may assume byinduction that Hom R p,q ( B p,q , D p,q ( µ )) ∼ = L p,q ( µ ). Applying Corollary 4.8 again,we deduce that L p,q ( µ ) is a quotient of E i L . So we get that L ∼ = L p,q ( λ ) byCorollary 4.4. (cid:3) Multiplicity-free version of level two Schur-Weyl duality. Continuewith p ≤ q . Let P p,q := T p,q ⊗ R p,q B p,q . (4.9)This is a ( G, K p,q )-bimodule, i.e. it is both a G -module and right K p,q -moduleso that the right action of K p,q is by G -module endomorphisms. Theorem 4.10. The homomorphism K p,q ∼ → End G ( P p,q ) op induced by the rightaction of K p,q on P p,q is an isomorphism. Moreover: (i) There is an isomorphism ζ : Hom G ( P p,q , ?) ∼ → Hom R p,q ( B p,q , ?) ◦ Hom G ( T p,q , ?) of functors from F to rep( K p,q ) . (ii) There is an isomorphism t : E i ◦ Hom G ( P p,q , ?) ∼ → Hom G ( P p,q , ?) ◦ E i of functors from F to rep( K p,q ) . HOVANOV’S DIAGRAM ALGEBRA IV 35 (iii) We have that Hom G ( P p,q , L ( λ )) ∼ = L p,q ( λ ) for each λ ∈ Λ ◦ p,q . (iv) As a G -module, P p,q decomposes as L λ ∈ Λ ◦ p,q P p,q e λ with P p,q e λ ∼ = P ( λ ) for each λ ∈ Λ ◦ p,q .Proof. We have natural isomorphismsEnd G ( P p,q ) op = Hom G ( T p,q ⊗ R p,q B p,q , T p,q ⊗ R p,q B p,q ) ∼ = Hom R p,q ( B p,q , Hom G ( T p,q , T p,q ⊗ R p,q B p,q )) ∼ = Hom R p,q ( B p,q , Hom G ( T p,q , T p,q ) ⊗ R p,q B p,q ) ∼ = Hom R p,q ( B p,q , R p,q ⊗ R p,q B p,q ) ∼ = End R p,q ( B p,q ) op ∼ = K p,q , using Corollary 4.6. This proves the first statement in the theorem.Then for (i), we use the natural isomorphismsHom G ( P p,q , M ) = Hom G ( T p,q ⊗ R p,q B p,q , M ) ∼ = Hom R p,q ( B p,q , Hom G ( T p,q , M )) . For (ii), we combine (i), Corollary 4.8 and Lemma 3.25; the isomorphism t isgiven explicitly by the natural transformation ζ − ◦ r ◦ s ◦ ζ . For (iii), useLemma 4.9 and the definition (3.19).Finally, consider (iv). The fact that P p,q = L λ ∈ Λ ◦ p,q P p,q e λ follows as theidempotents { e λ | λ ∈ Λ ◦ p,q } sum to the identity in K p,q . Note as B p,q is projectiveas a left R p,q -module, it is a summand of a direct sum of copies of R p,q as a leftmodule. Hence as a G -module P p,q is a summand of a direct sum of copies of T p,q . Applying Corollary 3.7, we deduce that the indecomposable summandsof P p,q as a G -module are all of the form P ( λ ) for various λ ∈ Λ ◦ p,q . Moreover,for any λ, µ ∈ Λ ◦ p,q , we have thatdim Hom G ( P p,q e λ , L ( µ )) = dim e λ Hom G ( P p,q , L ( µ )) = dim e λ L p,q ( µ ) = δ λ,µ , using (iii) and the definition of L p,q ( µ ). This completes the proof. (cid:3) Direct limits In this section we complete the proof of Theorem 1.1 by taking a limit as p → −∞ and q → ∞ . Various embeddings. In this subsection we fix p ′ ≤ p ≤ q ≤ q ′ such that either p ′ = p − q ′ = q or p ′ = p and q ′ = q + 1. By definition, the algebra K p,q is equal to the subring e p,q K p ′ ,q ′ e p,q of K p ′ ,q ′ . So P p ′ ,q ′ e p,q is a ( G, K p,q )-bimodule. The goal is to construct an isomorphism π p ′ ,q ′ p,q : P p,q ∼ → P p ′ ,q ′ e p,q .Throughout the subsection, we set i := (cid:26) ( p ′ , p ′ − , . . . , p ′ − m + 1) if p ′ = p − q ′ , q ′ + 1 , . . . , q ′ + n − 1) if q ′ = q + 1. (5.1)We have that i ∈ ( I p ′ ,q ′ ) c where c := m if p ′ = p − c := n if q ′ = q + 1.Introduce the idempotent ξ i := X d ≥ ξ i ; d ∈ R p ′ ,q ′ where ξ i ; d = X j ∈ ( I p,q ) d e ( i + j ) ∈ R p ′ ,q ′ c + d , (5.2)writing i + j for the sequence ( i , . . . , i c , j , . . . , j d ). The following lemma ex-plains how to identify R p,q with the subring ξ i R p ′ ,q ′ ξ i of R p ′ ,q ′ . Lemma 5.1. Let t = t c · · · t be the composite matching and Γ = Γ c · · · Γ bethe block sequence associated to the ( p ′ , q ′ ) -admissible sequence i from (5.1).Let γ = γ c · · · γ be the unique sequence of weights with γ r ∈ Γ r for each r ; inparticular, γ = λ p ′ ,q ′ and γ c = λ p,q . Then there is a unital algebra isomorphism ρ p ′ ,q ′ p,q : R p,q ∼ → ξ i R p ′ ,q ′ ξ i defined on the basis of oriented stretched circle diagrams by setting ρ p ′ ,q ′ p,q ( | s ∗ [ τ ∗ ] ≀ r [ σ ] | ) := | t ∗ [ γ ∗ ] ≀ s ∗ [ τ ∗ ] ≀ r [ σ ] ≀ t [ γ ] | , i.e. we glue γ t ∗ γ · · · γ c − t ∗ c onto the bottom and t c γ c − · · · γ t γ onto thetop of the given diagram s ∗ [ τ ∗ ] ≀ r [ σ ] . Moreover, writing ρ p ′ ,q ′ p,q = P d ≥ ρ d forisomorphisms ρ d : R p,qd ∼ → ξ i ; d R p ′ ,q ′ c + d ξ i ; d , the following two properties hold. (i) On the Khovanov-Lauda-Rouquier generators of R p,qd , we have that ρ d ( e ( j )) = e ( i + j ) , ρ d ( ψ r ) = ξ i ; d ψ c + r , ρ d ( y s ) = ξ i ; d y c + s , for j ∈ ( I p,q ) d , ≤ r < d and ≤ s ≤ d . (ii) On the Hecke generators of R p,qd , we have that ρ d ( s r ) = ξ i ; d s c + r , ρ d ( x s ) = ξ i ; d x c + s , for ≤ r < d and ≤ s ≤ d .Proof. The existence of the isomorphism ρ p ′ ,q ′ p,q is a consequence of the dia-grammatic description of the algebras R p,q and ξ i R p ′ ,q ′ ξ i . One first checks byinspecting bases that the given linear map is a vector space isomorphism, thenthat it preserves multiplication. The latter is obvious because we have justadded some extra line segments all oriented ∧ at the top and bottom of thediagram.To check (i), it follows from (3.12) and the definitions just before (3.15) and(3.14) that ρ d ( e ( j )) = e ( i + j ), ρ d ( ¯ ψ r ) = ξ i ; d ¯ ψ c + r and ρ d (¯ y s ) = ξ i ; d ¯ y c + s . Itremains to show that the signs (3.13) involved in passing from ¯ ψ to ψ and from¯ y to y match up correctly, which amounts to the observation that σ rp,q ( j ) = σ c + rp ′ ,q ′ ( i + j ) for j ∈ ( I p,q ) d and 1 ≤ r ≤ d . This follows from the identitymin( p, j r ) + min( q, j r ) − p = min( p ′ , j r ) + min( q ′ , j r ) − p ′ − δ i ,j r − · · · − δ i c ,j r , which we leave as an exercise for the reader. Then (ii) follows from (i) andTheorem 3.12. (cid:3) We can make a very similar construction at the level of the bimodule B p,q .In the following lemma, we view ξ i B p ′ ,q ′ e p,q as an ( R p,q , K p,q )-bimodule, wherethe left R p,q -module structure is defined via the isomorphism from Lemma 5.1. Lemma 5.2. Let t and γ be as in Lemma 5.1. There is an isomorphism of ( R p,q , K p,q ) -bimodules β p ′ ,q ′ p,q : B p,q ∼ → ξ i B p ′ ,q ′ e p,q defined on the basis of oriented upper-stretched circle diagrams by setting β p ′ ,q ′ p,q ( | u ∗ [ δ ∗ ] b )) := | t ∗ [ γ ∗ ] ≀ u ∗ [ δ ∗ ] b ) . HOVANOV’S DIAGRAM ALGEBRA IV 37 Moreover, the map κ p ′ ,q ′ p,q : R p ′ ,q ′ ξ i ⊗ R p,q B p,q → B p ′ ,q ′ e p,q , x ⊗ b xβ p ′ ,q ′ p,q ( b ) is an isomorphism of ( R p ′ ,q ′ , K p,q ) -bimodules.Proof. The fact that β p ′ ,q ′ p,q is an isomorphism of vector spaces follows by consid-ering the explicit diagram bases, and it is obviously a bimodule homomorphism.To deduce the final part of the lemma, it remains to show that the natural mul-tiplication map R p ′ ,q ′ ξ i ⊗ ξ i R p ′ ,q ′ ξ i ξ i B p ′ ,q ′ e p,q → B p ′ ,q ′ e p,q is an isomorphism. For this, we argue like in the proof of Lemma 4.3, startingfrom the trivial observation that the multiplication map R p ′ ,q ′ ξ i ⊗ ξ i R p ′ ,q ′ ξ i ξ i R p ′ ,q ′ ξ i → R p ′ ,q ′ ξ i is an isomorphism. Thus, we are reduced to showing that all the indecom-posable summands of B p ′ ,q ′ e p,q are also summands of R p ′ ,q ′ ξ i as left R p ′ ,q ′ -modules. By Corollary 4.6 and Lemma 4.9, we know that the indecomposablesummands of B p ′ ,q ′ e p,q are the projective covers of the irreducible R p ′ ,q ′ -modules { D p ′ ,q ′ ( λ ) | λ ∈ Λ ◦ p,q } . Since R p ′ ,q ′ ξ i is projective, it just remains to check thatHom R p ′ ,q ′ ( R p ′ ,q ′ ξ i , D p ′ ,q ′ ( λ )) = ξ i D p ′ ,q ′ ( λ ) = { } for λ ∈ Λ ◦ p,q . By Lemma 2.6, we can find d ≥ j ∈ ( I p,q ) d suchthat λ p,q j → · · · j d → λ is a path in the crystal graph. As λ p ′ ,q ′ i → · · · i c → λ p,q is apath in the crystal graph too, we get by repeated application of Corollary 3.26that E i · · · E i c E j · · · E j d D p ′ ,q ′ ( λ ) = { } . By the definition of the i -restrictionfunctors, this means that e ( i + j ) D p ′ ,q ′ ( λ ) = { } . Since ξ i e ( i + j ) = e ( i + j ),this implies that ξ i D p ′ ,q ′ ( λ ) = { } too. (cid:3) Next we explain how to identify T p,q with T p ′ ,q ′ ξ i . Lemma 5.3. There exists a (unique up to scalars) G -module isomorphism τ p ′ ,q ′ p,q : T p,q ∼ → T p ′ ,q ′ ξ i such that τ p ′ ,q ′ p,q = P d ≥ τ d for isomorphisms τ d : T p,qd ∼ → T p ′ ,q ′ c + d ξ i ; d with τ d +1 = P k ∈ I p,q F k ( τ d ) for each d ≥ . Moreover, τ p ′ ,q ′ p,q is a homomorphism of right R p,q -modules, i.e. it is a ( G, R p,q ) -bimodule isomorphism, where we are viewing T p ′ ,q ′ ξ i as a right R p,q -module via the isomorphism from Lemma 5.1.Proof. We first construct the map τ . Recall that T p,q = V ( λ p,q ) and T p ′ ,q ′ c ξ i ;0 =( V ( λ p ′ ,q ′ ) ⊗ V ⊗ c ) e ( i ) = F i V ( λ p ′ ,q ′ ). By Lemma 2.12 we have that F i V ( λ p ′ ,q ′ ) ∼ = V ( λ p,q ); only the analogues of the statements from (i) and (iii) of Lemma 2.4are needed to see this. So we can pick a G -module isomorphism τ : T p,q = V ( λ p,q ) ∼ → F i V ( λ p ′ ,q ′ ) = T p ′ ,q ′ c ξ i ;0 . This map is unique up to a scalar.Now we inductively define the higher τ d ’s. Note as T p,qd = L j ∈ ( I p,q ) d F j V ( λ p,q )that T p,qd +1 = L k ∈ I p,q F k T p,qd . Similarly T p ′ ,q ′ c + d +1 ξ i ; d +1 = L k ∈ I p,q F k ( T p ′ ,q ′ c + d ξ i ; d ). So given a G -module isomorphism τ d : T p,qd ∼ → T p ′ ,q ′ c + d ξ i ; d for some d ≥ 0, we geta G -module isomorphism τ d +1 : T p,qd +1 ∼ → T p ′ ,q ′ c + d +1 ξ i ; d +1 on applying the functor L k ∈ I p,q F k . Starting from the map τ from the previous paragraph, we obtainisomorphisms τ d for every d ≥ τ p ′ ,q ′ p,q := P d ≥ τ d , toget the desired G -module isomorphism T p,q ∼ → T p ′ ,q ′ ξ i .It remains to check that each τ d is a homomorphism of right R p,qd -modules,viewing T p ′ ,q ′ c + d ξ i ; d as a right R p,qd -module via the isomorphism ρ d from Lemma 5.1.Because τ is a G -module homomorphism, the map V ( λ p,q ) ⊗ V ⊗ d → V ( λ p ′ ,q ′ ) ⊗ V ⊗ ( c + d ) , u ⊗ v τ ( u ) ⊗ v ( u ∈ V ( λ p,q ) , v ∈ V ⊗ d )intertwines the action of x s ∈ H d with x c + s ∈ H c + d . It obviously intertwinesthe action of each s r ∈ H d with s c + r ∈ H c + d . From this and the definition of τ d , we deduce that τ d intertwines the actions of s r , x s ∈ H d on T p,qd with theactions of s c + r , x c + s ∈ H c + d on T p ′ ,q ′ c + d ξ i ; d . So we are done by the description of ρ d from Lemma 5.1(ii). (cid:3) Recall finally the spaces P p,q = T p,q ⊗ R p,q B p,q from (4.9). Theorem 5.4. There is a unique (up to scalars) ( G, K p,q ) -bimodule isomor-phism π p ′ ,q ′ p,q : P p,q ∼ → P p ′ ,q ′ e p,q such that π p ′ ,q ′ p,q ( v ⊗ b ) = τ p ′ ,q ′ p,q ( v ) ⊗ β p ′ ,q ′ p,q ( b ) for v ∈ T p,q , b ∈ B p,q and somechoice of the isomorphism τ p ′ ,q ′ p,q from Lemma 5.3.Proof. Recalling the isomorphism κ p ′ ,q ′ p,q from Lemma 5.2, we define π p ′ ,q ′ p,q to bethe composition of the following ( G, K p,q )-bimodule isomorphisms: T p,q ⊗ R p,q B p,q τ p ′ ,q ′ p,q ⊗ id −→ T p ′ ,q ′ ξ i ⊗ R p,q B p,q ≡ T p ′ ,q ′ ⊗ R p ′ ,q ′ R p ′ ,q ′ ξ i ⊗ R p,q B p,q id ⊗ κ p ′ ,q ′ p,q −→ T p ′ ,q ′ ⊗ R p ′ ,q ′ B p ′ ,q ′ e p,q . It remains to observe that π p ′ ,q ′ p,q ( v ⊗ b ) = τ p ′ ,q ′ p,q ( v ) ⊗ β p ′ ,q ′ p,q ( b ), which follows fromthe definition of κ p ′ ,q ′ p,q . (cid:3) Compatibility of embeddings. Now we explain how to glue the isomor-phisms π p,q +1 p,q and π p − ,qp,q from the previous subsection together in a consistentway to obtain a compatible system of isomorphisms π p ′ ,q ′ p,q : P p,q ∼ → P p ′ ,q ′ e p,q forevery p ′ ≤ p ≤ q ≤ q ′ . The following lemma is the key ingredient making thispossible. Lemma 5.5. Let p ≤ q be fixed. Given a choice of three out of the four maps { π p,q +1 p,q , π p − ,qp,q , π p − ,q +1 p,q +1 , π p − ,q +1 p − ,q } from Theorem 5.4, there is a unique way to choose the fourth one so that π p − ,q +1 p,q +1 ◦ π p,q +1 p,q = π p − ,q +1 p − ,q ◦ π p − ,qp,q . HOVANOV’S DIAGRAM ALGEBRA IV 39 Proof. We show equivalently given a choice of all four maps that there is a(necessarily unique) scalar z ∈ F such that π p − ,q +1 p,q +1 ◦ π p,q +1 p,q = zπ p − ,q +1 p − ,q ◦ π p − ,qp,q . To see this, let h := ( p − , p − , . . . , p − m ) and i := ( q + 1 , q + 2 , . . . , q + n ).Let ψ := ( ψ m ψ m +1 · · · ψ m + n − ) · · · ( ψ ψ · · · ψ n +1 )( ψ ψ · · · ψ n ) ,ψ ′ := ( ψ n · · · ψ ψ )( ψ n +1 · · · ψ ψ ) · · · ( ψ m + n − · · · ψ m +1 ψ m ) . It is easy to see from the defining relations between the Khovanov-Lauda-Rouquier generators from [BS3, (6.8)–(6.16)] that ψξ i + h = ξ h + i ψ , ξ i + h ψ ′ = ψ ′ ξ h + i , and ψ ′ ψξ i + h = ξ i + h in R p − ,q +1 .Now we claim that there exists a scalar z ∈ F such that the following twodiagrams commute: B p,qβ p − ,qp,q ւ ց β p,q +1 p,q ξ h B p − ,q e p,q ξ i B p,q +1 e p,qβ p − ,q +1 p − ,q y y β p − ,q +1 p,q +1 ξ i + h B p − ,q +1 e p,q ∼ −−−−→ L ψ ξ h + i B p − ,q +1 e p,q , (5.3) T p,qτ p − ,qp,q ւ ց τ p,q +1 p,q T p − ,q ξ h T p,q +1 ξ i τ p − ,q +1 p − ,q y y τ p − ,q +1 p,q +1 T p − ,q +1 ξ i + h ∼ −−−−→ R zψ ′ T p − ,q +1 ξ h + i , (5.4)where L ψ ( b ) := ψb and R zψ ′ ( v ) := zvψ ′ . Given the claim and recallingLemma 5.4, we get for any v ⊗ b ∈ T p,q ⊗ R p,q B p,q that π p − ,q +1 p,q +1 ( π p,q +1 p,q ( v ⊗ b )) = τ p − ,q +1 p,q +1 ( τ p,q +1 p,q ( v )) ⊗ β p − ,q +1 p,q +1 ( β p,q +1 p,q ( b ))= zτ p − ,q +1 p − ,q ( τ p − ,qp,q ( v )) ψ ′ ⊗ ψβ p − ,q +1 p − ,q ( β p − ,qp,q ( b ))= zτ p − ,q +1 p − ,q ( τ p − ,qp,q ( v )) ⊗ ψ ′ ψβ p − ,q +1 p − ,q ( β p − ,qp,q ( b ))= zτ p − ,q +1 p − ,q ( τ p − ,qp,q ( v )) ⊗ β p − ,q +1 p − ,q ( β p − ,qp,q ( b ))= zπ p − ,q +1 p − ,q ( π p − ,qp,q ( v ⊗ b )) . So the lemma follows from the claim.To prove the claim, consider first the diagram (5.3). The point for thisis that all the ψ r ’s in the element ψ are acting successively on the left on ξ i + h B p − ,q +1 e p,q as a sequence of height moves in the sense of [BS3, § τ d ’s in Lemma 5.3 to checking just that the diagram commutes on restriction to T p,q = V ( λ p,q ). In that case, both of T p − ,q +1 m + n ξ i + h and T p − ,q +1 m + n ξ h + i are isomorphic to V ( λ p,q ), and the map definedby right multiplication by ψ ′ is a non-zero isomorphism. So the diagram mustcommute up to a scalar as End G ( V ( λ p,q )) is one dimensional. (cid:3) Theorem 5.6. We can choose ( G, K p,q ) -bimodule isomorphisms π p ′ ,q ′ p,q : P p,q ∼ → P p ′ ,q ′ e p,q for all p ′ ≤ p ≤ q ≤ q ′ , in such a way that π p ′′ ,q ′′ p,q = π p ′′ ,q ′′ p ′ ,q ′ ◦ π p ′ ,q ′ p,q whenever p ′′ ≤ p ′ ≤ p ≤ q ≤ q ′ ≤ q ′′ .Proof. First of all we make arbitrary choices for the maps π p,q +1 p,q from Theo-rem 5.4 for all p ≤ q . Also we make arbitrary choices for the maps π p − ,pp,p fromTheorem 5.4 for all p . Then we repeatedly apply Lemma 5.5 , proceeding byinduction on ( q − p ), to get maps π p − ,qp,q so that the following local relationholds π p − ,q +1 p,q +1 ◦ π p,q +1 p,q = π p − ,q +1 p − ,q ◦ π p − ,qp,q for all p ≤ q . Finally we define the maps π p ′ ,q ′ p,q in general by setting π p ′ ,q ′ p,q := π p ′ ,q ′ p ′ +1 ,q ′ ◦ · · · ◦ π p − ,q ′ p,q ′ π p,q ′ p,q ′ − ◦ · · · ◦ π p,q +1 p,q . The equality π p ′′ ,q ′′ p,q = π p ′′ ,q ′′ p ′ ,q ′ ◦ π p ′ ,q ′ p,q follows from this definition and the local relation. (cid:3) Proof of the main theorem. Consider the directed set { ( p, q ) | p ≤ q } where( p, q ) → ( p ′ , q ′ ) if p ′ ≤ p ≤ q ≤ q ′ . By Theorem 5.6, it is possible to choosea direct system { π p ′ ,q ′ p,q : P p,q → P p ′ ,q ′ e p,q } of ( G, K p,q )-bimodule isomorphismsfor every ( p, q ) → ( p ′ , q ′ ). Let P := lim −→ P p,q (5.5)be the corresponding direct limit taken in the category of all G -modules, anddenote the canonical inclusion of each P p,q into P by ϕ p,q . We make P into alocally unital right K -module as follows. Take x ∈ K and v ∈ P . RecallingLemma 4.1, we can choose p ≤ q so that x = e p,q xe p,q and v = ϕ p,q ( v p,q ) forsome v p,q ∈ P p,q . Then set vx := ϕ p,q ( v p,q x ). Remark 5.7. Note that P is independent of the particular choice of the maps { π p ′ ,q ′ p,q } in the sense that if ¯ P = lim −→ P p,q is another such direct limit taken withrespect to maps { ¯ π p ′ ,q ′ p,q } , then there is a unique bimodule isomorphism P ∼ → ¯ P such that ϕ p,q ( v ) ¯ ϕ p,q ( v ) for all v ∈ P p,q and p ≤ q .Roughly speaking, the following lemma shows that P is a minimal projectivegenerator for the category F (except that as P is not finite dimensional it isnot actually an object in the category). Lemma 5.8. As a G -module, we have that P = L λ ∈ Λ P e λ with P e λ ∼ = P ( λ ) for each λ ∈ Λ .Proof. The first part of the lemma is immediate because P is a locally unitalright K -module. To show that P e λ ∼ = P ( λ ), we have by the above definitionsthat P e λ = lim −→ ( P p,q e λ ) where the direct limit is taken over all p ≤ q with HOVANOV’S DIAGRAM ALGEBRA IV 41 λ ∈ Λ ◦ p,q (so that e λ ∈ K p,q ). Each P p,q e λ is isomorphic to P ( λ ) by Theo-rem 4.10(iv). Hence the direct limit is isomorphic to P ( λ ) too. (cid:3) Now we want to identify the algebra K with the endomorphism algebra of P .A little care is needed here as P is an infinite direct sum. So for any G -module M , we let Hom finG ( P, M ) := M λ ∈ Λ Hom G ( P e λ , M ) ⊆ Hom G ( P, M ) , (5.6)which is the locally finite part of Hom G ( P, M ). Note if M is finite dimensionalthat Hom finG ( P, M ) = Hom G ( P, M ). In particular, we denote Hom finG ( P, P ) byEnd finG ( P ) and write End finG ( P ) op for the opposite algebra, which acts naturallyon the right on P by G -module endomorphisms. Lemma 5.9. The right action of K on P defined above induces an algebraisomorphism K ∼ → End finG ( P ) op .Proof. We need to show that right multiplication induces a vector space iso-morphism e λ K ∼ → Hom G ( P e λ , P ) for each λ ∈ Λ. By definition, the right handspace is Hom G ( P e λ , [ P e p,q ) = [ Hom G ( P e λ , P e p,q )where we can take the union just over p ≤ q with λ ∈ Λ ◦ p,q . As P e λ = ϕ p,q ( P p,q e λ ) and P e p,q = ϕ p,q ( P p,q ) for all such p ≤ q , the first statementfrom Theorem 4.10 implies that right multiplication induces an isomorphism e λ Ke p,q ∼ → Hom G ( P e λ , P e p,q ). Taking the union and recalling Lemma 4.1, wededuce that we do get an isomorphism e λ K ∼ → S Hom G ( P e λ , P e p,q ). (cid:3) Finally we record the following variation on a basic fact. Lemma 5.10. Let B be a G -module that is also a locally unital right K -module,such that the action of K on B is by G -module endomorphisms. Let M be anyfinite dimensional left K -module and assume that B ⊗ K M is finite dimensional.Then there is a natural G -module isomorphism Hom finG ( P, B ) ⊗ K M → Hom G ( P, B ⊗ K M ) sending f ⊗ m to the homomorphism v f ( v ) ⊗ m .Proof. It suffices to show that Hom G ( P e λ , B ) ⊗ K M ∼ = Hom G ( P e λ , B ⊗ K M )for each λ ∈ Λ, which is well known. (cid:3) Now we can prove the main result of the article, which is essentially Theo-rem 1.1 from the introduction with the functor E there constructed explicitly.The proof is a rather standard consequence of the last three lemmas, but weinclude some details since we are in a slightly unusual locally finite setting. Theorem 5.11. The functors Hom G ( P, ?) : F → rep( K ) , P ⊗ K ? : rep( K ) → F are quasi-inverse equivalences of categories. Moreover P ⊗ K P ( λ ) ∼ = P ( λ ) foreach λ ∈ Λ . Proof. Note using Lemma 5.8 that both the functors map finite dimensionalmodules to finite dimensional modules, so the first statement makes sense.Lemmas 5.10 and 5.9 yield a natural isomorphismHom G ( P, P ⊗ K M ) ∼ → Hom finG ( P, P ) ⊗ K M ∼ = K ⊗ K M ≡ M for any M ∈ rep( K ). Thus Hom G ( P, ?) ◦ P ⊗ K ? ∼ = Id rep( K ) . Conversely, to showthat P ⊗ K ? ◦ Hom G ( P, ?) ∼ = Id F , we have a natural homomorphism P ⊗ K Hom G ( P, N ) → N, v ⊗ f f ( v )for every N ∈ F . Because of Lemma 5.8 this map is surjective. To show that itis injective too, denote its kernel by U . Applying the exact functor Hom G ( P, ?),we get a short exact sequence0 → Hom G ( P, U ) → Hom G ( P, P ⊗ K Hom G ( P, N )) → Hom G ( P, N ) → . By the fact established just before, the middle space here is isomorphic toHom G ( P, N ), so the right hand map is an isomorphism. Hence Hom G ( P, U ) = { } , which implies that U = { } . So our natural transformation is an isomor-phism, and we have established the equivalence of categories. Moreover, P ⊗ K P ( λ ) = P ⊗ K Ke λ ≡ P e λ ∼ = P ( λ )by Lemma 5.8. (cid:3) Theorem 1.1 from the introduction is a consequence of Theorem 5.11, taking E := Hom G ( P, ?). We have already proved that E P ( λ ) ∼ = P ( λ ), which immedi-ately implies that E L ( λ ) ∼ = L ( λ ). The fact that E V ( λ ) ∼ = V ( λ ) follows becauseboth the categories F and rep( K ) are highest weight categories in which themodules {V ( λ ) } and { V ( λ ) } give the standard modules; see Theorem 2.15 forthe former and [BS1, Theorem 5.3] for the latter fact. Identification of special projective functors. Finally we discuss brieflyhow to relate the special projective functors on the two sides of our equivalenceof categories. Theorem 5.12. For each i ∈ I , we have that E i ∼ = Hom G ( P, ?) ◦ E i ◦ P ⊗ K ? , F i ∼ = Hom G ( P, ?) ◦ F i ◦ P ⊗ K ? as endofunctors of rep( K ) .Proof. Since F i is left adjoint to E i and F i is left adjoint to E i , the secondisomorphism is a consequence of the first, by unicity of adjoints. To prove thefirst, we note using Lemma 5.10 that there are natural isomorphismsHom G ( P, E i ( P ⊗ K M )) ∼ = Hom G ( P, ( E i P ) ⊗ K M ) ∼ = Hom finG ( P, E i P ) ⊗ K M for any M ∈ rep( K ). Hence it suffices to show that Hom finG ( P, E i P ) ∼ = e E i as( K, K )-bimodules. For this, we just sketch how to construct the appropriatemap, leaving details to the reader. Take any λ ∈ Λ and any p ≤ q so that weactually have λ ∈ Λ ◦ p,q . When applied to the module P p,q , the natural isomor-phism from Theorem 4.10(ii) produces a ( K p,q , K p,q )-bimodule isomorphism ε p,q : e E p,qi ∼ → Hom G ( P p,q , E i P p,q ) . HOVANOV’S DIAGRAM ALGEBRA IV 43 Restricting this to e λ e E p,qi = e λ e E i e p,q and using ϕ p,q to identify P p,q with P e p,q ,we get from this a vector space isomorphism ε p,q : e λ e E i e p,q ∼ → e λ Hom G ( P e p,q , E i P e p,q ) = Hom G ( P e λ , E i P e p,q ) . Now one checks for p ′ ≤ p ≤ q ≤ q ′ that ε p,q ( v ) = ε p ′ ,q ′ ( v ) for all v ∈ e λ e E i e p,q ;it suffices to do this in the cases ( p ′ , q ′ ) = ( p − , q ) or ( p, q + 1). Hence it makessense to take the union over all p ≤ q to get an isomorphism ε : e λ e E i ∼ → Hom G ( P e λ , E i P ) . Taking the direct sum of these maps over all λ ∈ Λ gives finally the desiredmap e E i ∼ → Hom finG ( P, E i P ). (cid:3) Index of notation G = GL ( m | n ) General linear supergroup 1 V, V ∗ Natural representation of G and its dual 2 B, T Standard Borel subgroup and maximal torus of G F = F ( m | n ) Half of the category of finite dimensional G -modules 2 X + ( T ) Dominant weights 2 L ( λ ) , V ( λ ) , P ( λ ) Irreducibles, standards and PIMs for G for λ ∈ X + ( T ) 11 E i , F i Special projective functors for G K = K ( m | n ) Generalised Khovanov algebra 8Λ = Λ( m | n ) Diagrammatic weights in bijection with X + ( T ) 2 L ( λ ) , V ( λ ) , P ( λ ) Irreducibles, standards and PIMs for K for λ ∈ Λ 8 E i , F i Special projective functors for K λ p,q Ground-state weight 4 H p,qd Cyclotomic Hecke algebra which acts on V ( λ p,q ) ⊗ V ⊗ d I + p,q Index set for pq -strip 11Λ p,q , Λ ◦ p,q Weights, weights of maximal defect in pq -strip 111 p,qd Central idempotent in H p,qd corresponding to pq -strip 19 T p,qd Tensor space ( V ( λ p,q ) ⊗ V ⊗ d )1 p,qd R p,qd Cyclotomic KLR algebra ∼ = 1 p,qd H p,qd ∼ = End G ( T p,qd ) op T p,q , R p,q Direct sums L d ≥ T p,qd and L d ≥ R p,qd K p,q Subring of K that is Morita equivalent to R p,q A p,q , B p,q Morita bimodules 32 P p,q Multiplicity-free projective module T p,q ⊗ R p,q B p,q P = lim −→ P p,q Canonical minimal projective generator for F References [Ba] H. 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