Canonical bases and higher representation theory

Transcription

1 Ben Webster 1 Department of Mathematcs Northeastern Unversty Boston, MA Emal: b.webster@neu.edu Abstract. We show that Lusztg s canoncal bass n the whole quantzed unversal envelopng algebra s gven by the classes of the ndecomposable 1-morphsms n a 2-category categorfyng the unversal envelopng algebra, when the assocated Le algebra s fnte type and smply laced. The 2-category s a varaton on those defned by Rouquer and Khovanov-Lauda descrbed n a recent paper of Cauts and Lauda. Furthermore, we ntroduce natural categores whose Grothendeck groups correspond to the tensor products of lowest and hghest weght ntegrable representatons. Ths s a natural generalzaton of a smlar constructon of categores for tensor products of hghest weght ntegrable representatons from the authors prevous work. More generally, we study the theory of bases arsng from ndecomposable obects n hgher representaton theory, whch we term orthodox and the overlap of ths theory wth the more classcal theory of canoncal bases. Contents 1. The 2-category U 6 2. The 2-category T Tensor product algebras Representaton categores and standard modules Orthodox bases Canoncal bases Dual canoncal bases 35 References 39 One of the consstent motvatons for the constructon of categorfcatons has been the accompanyng appearance of canoncal bases n the orgnal obect under consderaton. At ts core, ths s a consequence of a very smple prncple: the ndecomposable obects of any Krull-Schmdt category gve a bass of ts splt Grothendeck group. Furthermore, any map between Grothendeck groups whch lfts to a functor must have postve nteger coeffcents n ths bass. 1 Supported by the NSF under Grant DMS and by the NSA under Grant H

2 Whle ths postvty s an appealng consequence, on ts own, t has trouble makng up for the dffculty of computng ths bass n many stuatons; for example, rreducble characters gve a bass of class functons on a fnte group on whch multplcaton has postve ntegral structure coeffcents, but fndng rreducble characters s stll very hard n general. On the other hand, n some examples, these bases have consderably more structure. We let V be a free Z[q, q 1 ]-module; a pre-canoncal structure on V s a choce of a bar nvoluton ψ: V V whch s Z[q, q 1 ]- ant-lnear, a sesqulnear nner product, : V V Z((q 1 )), for whch ψ s flpuntary, u, v = ψ(v), ψ(u). a standard bass a c wth partally ordered ndex set (C, <) such that ψ(a c ) a c + Z[q, q 1 ] a c. We call a bass {b c } of V canoncal f I. each vector b c n the bass s nvarant under ψ. II. each vector b c n the bass s n the set a c + c <c Z[q, q 1 ] a c. III. the vectors b c are almost orthonormal n the sense that c <c b c, b c δ c,c + q 1 Z[[q 1 ]]. A well-trodden argument shows that such a bass s unque f t exsts, but showng exstence s generally qute dffcult. Whle we know of nowhere n the lterature where ths defnton s made n ths generalty, there are many examples. In each case, we wll leave the detals of the pre-canoncal structure to the references: Kazhdan and Lusztg showed that the Hecke algebra of a Weyl group has a canoncal bass [KL79], now usually called the Kazhdan-Lusztg bass. Lusztg showed that the smple ntegrable representatons of quantzed unversal envelopng algebras of Kac-Moody algebras as well as a small modfcaton U of the algebras themselves have canoncal bases [Lus93]. These also appeared n the work of Kashwara as global crystal bases. In fnte type, the tensor product of smple representatons also carres a natural canoncal bass [Lus93]; n the specal case of a tensor product of hghest and lowest weght representatons, ths works for nfnte type Kac- Moody algebras as well [Lus92]. Lascoux, Leclerc and Thbon [LLT96] show that a level 1 Fock space representaton of ŝl n carres a canoncal bass. Ths was extended to hgher level twsted Fock spaces by Uglov [Ugl00]; Brundan and Kleshchev [BK09a] showed that 2

3 Ben Webster tensor products of level 1 Fock spaces also have a canoncal bass arsng as a lmt of Uglov s. Pre-canoncal structures arse most naturally from categorfcatons: the nvoluton ψ s the decategorfcaton of a dualty functor, the standard bass s the decategorfcaton of some set of easly located obects where each contans exactly one ndecomposable summand not found n smaller ones, and the parng s gven by the graded Euler parng [M], [N] = dm q Mor(M, N). A canoncal bass wll arse when each ndecomposable has a choce of gradng shft n whch t s self-dual and n ths choce of gradng, there are no negatve degree maps between ndecomposables and only scalar multplcaton n degree 0; n ths case, the categorfcaton s sad to be mxed. Indeed, all of the bases lsted above have close tes to categorfcatons: The Kazhdan-Lusztg bass arses from the categorfcaton of the Hecke algebra by B B-equvarant mxed sheaves on G, the assocated algebrac group [Spr82]; alternatvely, there s an equvalent approach usng ndecomposable Soergel bmodules [Soe92]. For sl n, the canoncal bass of a tensor product of fundamental representatons corresponds to the proectve (or tltng, dependng on conventons) obects n a parabolc category O, equpped wth ts Koszul gradng [Sus, Th. 6]. For ŝl n, the canoncal bass on a level 1 Fock space arses from a graded verson of the q-schur algebra (for q an nth root of unty) [Ar09]; ths was recently extended to tensor products of level 1 Fock spaces by the author and Catharna Stroppel [SW] usng the cyclotomc q-schur algebras of Dpper, James and Mathas [DJM98]. The case of general hgher level Fock spaces n the sense of Uglov [Ugl00] remans more open; a conecture of Rouquer relates ths canoncal bass to category O for symplectc reflecton algebras of cyclotomc type [Rou08, 6.5]. There s also a dagrammatc categorfcaton of these spaces, whch the author wll descrbe n [Webd]. For sl 2, the ndecomposable obects of U match Lusztg s canoncal bass by work of Lauda [Lau10, 9.12]. The am of ths paper s to gve a coherent account of the remanng tems on our lst of canoncal bases, those arsng n quantzed unversal envelopng algebras and ther representatons usng hgher representaton theory, as developed by Rouquer, Khovanov, Lauda and others. We buld on very mportant results of Vasserot-Varagnolo [VV11, 4.5] to show: Theorem A (Theorems 6.8 & 6.11) If g s fnte type and smply-laced (that s, of ADE type), then the canoncal bass of the modfed quantzed unversal envelopng 3

4 algebra U or an arbtrary tensor product of fnte dmensonal representatons concdes wth the classes of ndecomposable obects n an approprate hgher analogue whose Grothendeck group s the abelan group n queston. If g s an arbtrary Kac-Moody algebra wth symmetrc Cartan matrx, the canoncal bass of a tensor product of hghest weght ntegrable representatons satsfes the same property. Unfortunately, n nfnte type, we can nether prove nor dsprove the concdence of the canoncal bass of U wth the classes of ndecomposables n the obvous canddate. The proof of Theorem A uses that hghest and lowest weght modules of U are the same n a very strong way. A general proof wll requre very dfferent technques, whch we hope can be suppled by the categorcal actons on quantzatons of quver varetes descrbed n [Weba]. At the moment, ths route s blocked by the lack of a fullness result whch s equvalent to Krwan surectvty for quver varetes; ths s a long-standng open problem, so untl t fnds a soluton, we cannot use ths approach. The careful reader wll note that n our account of canoncal bases, we only specfed that Lusztg had defned a canoncal bass for arbtrary tensor products of smples n fnte type, whereas above we have no such restrcton on type. The categorfcatons we dscuss allow us to defne a bar nvoluton on arbtrary tensor products of hghest and lowest weght representatons, whch concdes wth Lusztg s n the cases where he has defned t. We wsh to consder the bases whch are canoncal wth respect to that bar nvoluton. We defne a pre-canoncal structure on an arbtrary tensor product of hghest and lowest weght representatons, but the technques n the proof of Theorem A do not suffce to prove that a canoncal bass exsts n ths case, let alone that such a bass arses from a categorfcaton. In the body of the paper, we defne the categorfcatons of tensor products of hghest and lowest weght representatons mentoned n Theorem A; these are generalzatons of the categorfcatons of hghest weght representatons defned by the author n [Webb]. The structure of ths categorfcaton gves the correspondng tensor product a natural pre-canoncal structure exactly as descrbed above. Also, the tensor products have bases arsng naturally from the ndecomposable obects (n the language of [Webb, Webc], these would be ndecomposable proectves), whch we call orthodox 2. We develop the basc theory of these bases. Most mportantly, we show that these bases always satsfy condtons I. and II. (even n non-symmetrc type) for our chosen pre-canoncal structure. However, t requres sgnfcant geometrc nput to prove that n some cases, the condton III. holds as well. Ths s provded by calculatons wth perverse sheaves that appear n [Webd], buldng on work of Vasserot and Varagnolo [VV11]. 2 The word orthodox comes from the Greek ὸρθός correct + δόξα belef ; t s a bass we can beleve n. 4

5 Ben Webster We also brefly dscuss the phenomenon of dual canoncal bases. One can nterpret ths as smply meanng the dual to the canoncal bass under the form,. However, t also reflects a dualty operaton on canoncal structures; we dscuss both ths operaton and ts nteracton wth postvty theorems for canoncal bases. Acknowledgements. I thank George Lusztg; wthout hs questonng, ths paper mght never have happened. I also thank Jon Brundan, Yqang L, Ivan Losev, Marco Mackaay and Catharna Stroppel for useful dscussons. Notaton. We let g be a symmetrzable Kac-Moody algebra. Consder the weght lattce Y(g) and root lattce X(g), and the smple roots α and coroots α. Let c = α (α ) be the entres of the Cartan matrx. We let, denote the symmetrzed nner product on Y(g), fxed by the fact that the shortest root has length 2 and 2 α, α, α = α (). As usual, we let 2d = α, α, and for Y(g), we let = α () = α, /d. Throughout the paper, we wll use = ( 1,..., l ) to denote an ordered l-tuple of domnant or ant-domnant weghts, and always use the notaton =. We let U q (g) denote the deformed unversal envelopng algebra of g; that s, the assocatve C(q)-algebra gven by generators E, F, K µ for and µ Y(g), subect to the relatons: ) K 0 = 1, K µ K µ = K µ+µ for all µ, µ Y(g), ) K µ E = q α (µ) E K µ for all µ Y(g), ) K µ F = q α (µ) F K µ for all µ Y(g), K v) E F F E = δ K, where K q d q d ± = K ±d α, v) For all ( 1) a E (a) E E (b) = 0 and a+b= c +1 a+b= c +1 ( 1) a F (a) F F (b) = 0. Ths s a Hopf algebra wth coproduct on Chevalley generators gven by (E ) = E 1 + K E (F ) = F K + 1 F and antpode on these generators defned by S(E ) = K E, S(F ) = F K We let U Z q (g) denote the Lusztg (dvded powers) ntegral form generated over Z[q, q 1 ] by En [n] q!, F n [n] q! for all ntegers n of ths quantum group. The ntegral form of the representaton of hghest weght f s domnant or lowest weght f s ant-domnant over ths quantum group wll be denoted by V Z, and V Z = VZ 1 Z[q,q 1 ] Z[q,q 1 ] V Z l. 5

6 We let V Z denote the reducton of VZ at q = The 2-category U In ths paper, our notaton bulds on that of Khovanov and Lauda, who gve a graphcal verson of the 2-quantum group, whch we denote U (leavng g understood). These constructons could also be rephrased n terms of Rouquer s descrpton and we have strven to make the paper readable followng ether [KL10] or [Rou]; however, t s most sensble for us to follow the 2-category defned by Cauts and Lauda [CL] whch s a varaton on both of these. The dfference between ths category and the categores defned by Rouquer n [Rou] s qute subtle; t concerns precsely whether the nverse to a partcular map s formally added, or mposed to be a partcular composton of other generators n the category. Most mportant for our purposes, the 2-category U receves a canoncal map from each of Rouquer s categores A and A, so a representaton of t s a representaton n Rouquer s sense as well. In fact, n [CL], Cauts and Lauda show that under very mld condtons, the converse also holds: an acton of U s equvalent to a categorcal g-acton n the weaker sense. Snce the constructon of these categores s rather complex, we gve a somewhat abbrevated descrpton. Before the defnton, we must fx a commutatve complete local rng k wth maxmal deal m, and fx a matrx of polynomals Q (u, v) for Γ (by conventon Q = 0) valued n k. Typcally, we wll be nterested n the case where k s ust a feld, but the case of the p-adc ntegers (or other completons of number felds) s very useful for nterpolatng between characterstc 0 and characterstc p behavor. We assume each polynomal s homogeneous of degree α, α = 2d c = 2d c when u s gven degree 2d and v degree 2d. We wll always assume that the leadng order of Q n u s c, and that Q (u, v) = Q (v, u). We let t = Q (1, 0); we always assume that these elements of k are unts. By conventon t = 1. (We should warn the reader, n [CL] ths scalar s allowed to be any non-zero number; we avoded ths n order to smplfy our relatons). Khovanov and Lauda s category s the choce Q = u c + v c. We defne a category U where an obect of ths category s a weght Y. a 1-morphsm µ s a formal sum of words n the symbols E and F where ranges over Γ of weght µ, E and F havng weghts ±α. In [Rou], the correspondng 1-morphsms are denoted E, F, but we use these for elements of U q (g). Composton s smply concatenaton of words. In fact, we wll take dempotent completon, and thus add a new 1-morphsm for every proecton from a 1-morphsm to tself (once we have added 2-morphsms). By conventon, F = F n F 1 f = ( 1,..., n ) (ths somewhat dyslexc conventon s desgned to match prevous work on cyclotomc quotents by 6

7 Ben Webster Khovanov-Lauda and others). In Khovanov and Lauda s graphcal calculus, ths 1-morphsm s represented by a sequence of dots on a horzontal lne labeled wth the sequence. We should warn the reader, ths conventon requres us to read our dagrams dfferently from the conventons of [Lau10, KL10, CL]; n our dagrammatc calculus, 1-morphsms pont from the left to the rght, not from the rght to the left as ndcated n [Lau10, 4]. Techncally, the 2-category U we defne s the 1-morphsm dual of Khovanov and Lauda s 2-category: the obects are the same, but the 1-morphsms are all reversed. The practcal mplcaton wll be that our relatons are the reflecton through a vertcal lne of Cauts and Lauda s (wthout changng the labelng of regons). 2-morphsms are a certan quotent of the k-span of certan mmersed orented 1-manfolds carryng an arbtrary number of dots whose boundary s gven by the doman sequence on the lne y = 1 and the target sequence on y = 0. We requre that any component begn and end at lke-colored elements of the 2 sequences, and that they be orented upward at an E and downward at an F. We wll descrbe ther relatons momentarly. We requre that these 1- manfolds satsfy the same genercty assumptons as proectons of tangles (no trple ponts or tangences), but ntersectons are not over- or under-crossngs; our dagrams are genunely planar. We consder these up to sotopy whch preserves ths genercty. We draw these 2-morphsms n the style of Khovanov-Lauda, by labelng the regons of the plane by the weghts (obects) that the 1-morphsms are actng on. By Morse theory, we can see that these are generated by a cup ɛ : E F or ɛ : F E and a cap ι : E F or ι : F E ɛ = + α ɛ = ι = ι = α + α α a crossng ψ : F F F F and a dot y : F F ψ = y = 7

8 Before wrtng the relatons, let us remnd the reader that these 2-morphsm spaces are actually graded; the degrees are gven by deg = α, α deg = α, α deg = α, α deg = α, α deg =, α d deg =, α d deg =, α d deg =, α d. The relatons satsfed by the 2-morphsms nclude: the cups and caps are the unts and counts of a baduncton. The morphsm y s cyclc, whereas the morphsm ψ s double rght dual to t /t ψ (see [CL] for more detals). Any bubble of negatve degree s zero, any bubble of degree 0 s equal to 1. We must add formal symbols called fake bubbles whch are bubbles labelled wth a negatve number of dots (these are explaned n [KL10, 3.1.1]); gven these, we have the nverson formula for bubbles, shown n Fgure k= 1 k k = { 1 = 2 0 > 2 Fgure 1. Bubble nverson relatons; all strands are colored wth α. 2 relatons connectng the crossng wth cups and caps, shown n Fgure 2. Oppostely orented crossngs of dfferently colored strands smply cancel, shown n Fgure 3. the endomorphsms of words only usng F (or by dualty only E s) satsfy the relatons of the quver Hecke algebra R, shown n Fgure 4. As n [KL10], we let U denote the 2-category where every Hom-category s replaced by ts dempotent completon; we note that snce every obect n U has a fntedmensonal degree 0 part of ts endomorphsm algebra, every Hom-category satsfes the Krull-Schmdt property. Ths 2-category s a categorfcaton of the unversal envelopng algebra s the sense that Theorem 1.1 ([Webb, 1.7-9]) The Grothendeck group of U s somorphc to U and ts graded Euler form s gven by Lusztg s nner product (, ) on U. 8

9 Ben Webster = a+b= 1 b a = a+b= 1 b a = + a+b+c= 1 c a b = + a+b+c= 1 c a b Fgure 2. Cross and cap relatons; all strands are colored wth α. By conventon, a negatve number of dots on a strand whch s not closed nto a bubble s 0. = t = t Fgure 3. The cancellaton of oppostely orented crossngs wth dfferent labels. 9

10 = unless = = + = + = 0 and = Q (y 1, y 2 ) = unless = k k k = + Q (y 3, y 2 ) Q (y 1, y 2 ) y 3 y 1 Fgure 4. The relatons of the quver Hecke algebra. These relatons are nsenstve to labelng of the plane. Ths theorem was frst conectured by Khovanov and Lauda [KL10] and proven by them n the specal case of sl n. Whle not explctly stated n ther paper, ths also follows easly from [CL, 8.1] whch was proved ndependently of the work above, relyng on the paper of Kang and Kashwara [KK] n ts stead. 10

11 Ben Webster We recall from [KL10, 3.3.2] that we have an nvoluton ψ: Mor(E 1 E m, E 1 E n ) Mor(E 1 E n, E 1 E m ) reflectng the dagrams of two morphsms through a horzontal lne and reversng orentaton. Ths extends to a 2-functor U U whch s covarant on 1-morphsms and contravarant on 2-morphsms, sendng E (k) E ( k), F (k) F ( k). Proposton 1.2 (Khovanov-Lauda [KL10, 3.28]) The 2-functor ψ categorfes the bar nvoluton of U q (g) (denoted by ψ n [KL10]). Ths nner product and nvoluton are part of the pre-canoncal structure used by Lusztg to defne the canoncal bass of U; the role of the standard bass can be played by a number of dfferent bases of U. Ours wll be one perhaps less elegant on the level of the quantum groups than the PBW bass defned va the brad acton used by Lusztg n [Lus90], but s easer to handle n the categorfcaton. Ths bass wll be defned usng strng parametrzatons of crystal elements. 2. The 2-category T In the next three sectons, we wll present a constructon of a categorfcaton of tensor products of hghest and lowest weght representatons. Almost all of the results whch appear have equvalents n the author s earler paper [Webb], and n most cases, the nature of the proofs s qute smlar. Frst, we present an auxlary category whch generalzes that presented n [Webb, 2.11]. We defne a 2-category T whch s the 2-category whose obects are weghts X(g) 1-morphsms from to µ are sequences of domnant and ant-domnant weghts and postve and negatve smple roots whch sum to µ, or alternatvely, a sequence of postve and negatve smple roots and sequence of domnant and ant-domnant weghts together wth a weakly ncreasng functon κ : [1, l] [0, n] ndcatng whch entry s mmedately to the rght of (wth 0 ndcatng t s at the far left). The operaton of composton s smply concatenaton. As n U, we keep our dyslexc conventon that the composton a b s the concatenaton (b, a). 2-morphsms are orented mmersed 1-manfolds n R 2 wth boundares on the lnes y = 0 and y = 1, wth each component colored red, blue or black each red component colored wth a domnant weght each blue component wth ant-domnant weght each black component wth a smple root red and blue components cannot ntersect red or blue components (ncludng themselves), but can ntersect black components 11

12 the y-component s always ncreasng along blue components and always decreasng along red black components are allowed to carry dots, red and blue are not modulo relatons we wll descrbe shortly. The two sequences that 2-morphsms go between are read off from the lnes y = 0 and y = 1 wth a red or blue component gvng ts label, and a black component gvng t label f orented upward, or mnus ts label f orented downward. The relatons satsfed by the 2-morphsms are: black strands satsfy the relatons of U, shown n Fgures 1, 2 and 4. = = µ µ µ µ Fgure 5. The cancellaton of oppostely orented crossngs wth dfferent labels; the horzontal reflectons of these dagrams are also relatons. Oppostely orented crossngs of dfferently labelled strands smply cancel, shown n Fgure 5. Ths ncludes crossngs of red/blue strands wth black ones. All black crossngs and dots can pass through red or blue lnes, wth a correcton term smlar to Khovanov and Lauda s (for the latter 3 relatons, we also nclude ther mrror mages), as shown n Fgure 6. The cost of a separatng smlarly orented red/blue and black lnes s addng = α () dots to the black strand as shown n Fgure 7. As wth U and U, we let T be the dempotent completon of the Hom-categores of T. Ths 2-category carres an obvous acton of U by horzontal composton on the rght and on the left. In ths capacty t categorfes the tensor algebra of U q (g) endowed wth the left and rght acton. Unfortunately, as usual wth a presentaton by generators and relatons, t s far from obvous that ths s true, or even whether the category has any non-zero obects. In order to show ths, we show that t acts on the representaton categores of cyclotomc quotents. For two cyclotomc quotents T and T wth also domnant, we have an obvous map ϕ : T T compatble wth the map from the quver Hecke algebra 12

13 Ben Webster = a+b 1= δ, b a = + b a a+b 1= δ, = = = = Fgure 6. Passng crossngs and dots through colored lnes. = = = = to each. Thus, we have functors Fgure 7. Separatng smlarly orented strands ϑ = res T T : V V ε = T T : V V of restrcton and extenson of scalars between these categores. 13

14 We now descrbe a number of natural transformatons of these functors whch we wll use to buld a representaton of the 2-category T. (a) Snce the map ϕ s compatble wth the map T T addng a black strand colored at the far rght, there s an somorphsm a : ϑ E E ϑ and (b) a dual somorphsm b : ε F F ε both gven by dentfyng the underlyng vector spaces. On the other hand ϑ F F ϑ and ε E E ε. (c) Instead, there s a natural map c : F ϑ ϑ F nduced by the surectve map to both from the nducton over R; alternatvely, ths can be descrbed as comng from the unt of the aduncton of rght aduncton of E to F through the maps Mor(d, E F ) Mor(ϑ, ϑ E F ) Mor(ϑ, E ϑ F ) Mor(F ϑ, ϑ F ). (d) There s a dual map d : ϑ F F ϑ whch arses from count of the left aduncton of F to E va the maps Mor(E F, d) Mor(ϑ E F, ϑ ) Mor(E ϑ F, ϑ ) Mor(ϑ F, F ϑ ). More explctly, ths map between functors must arse from a map of T T bmodules from T thought of as a left T -module by the rng map ν addng a new strand of color, and as a rght T -module usng the map φ to T T T where tensor product s over the maps ν : T T and φ, endowed wth the obvous bmodule structure. Each of these s a quotent of the quver Hecke algebra R, thought of as a bmodule by the usual rght acton, and the left acton by ν R. However, ths quotent realzaton does not nduce a map; recall that there s a bmodule automorphsm y addng a dot to the new strand added by ν. The map d s nduced by the map y ( ) : R R snce ths map ntertwnes the operaton of closure n a plane labeled µ wth that n a plane labeled µ +. (g) Smlarly, there are maps g : ε E E ε arsng from the count of the frst aduncton by Mor(F E, d) Mor(ε F E, ε ) Mor(F ε E, ε ) Mor(ε E, E ε ) and (h) h : E ε ε E arsng from the unt of the second by Mor(d, F E ) Mor(ε, ε F E ) Mor(ε, F ε E ) Mor(E ε, ε E ). More explctly, g and h arse from the same bmodule maps as c and d after applyng vertcal reflecton to both the bmodules and algebras, n order to obtan a T T bmodule nstead. 14

15 Ben Webster Theorem 2.1 The category T acts on the drect sum X + V sendng: the acton of black lnes (smple roots) wth the functors E and F defned n [Webb, 1], the acton of a red lne labeled wth to the restrcton functor ϑ +µ µ, and the acton of a blue lne labeled wth to the nducton functor ε +µ µ (f µ s not domnant, ths functor s 0). On the level of morphsms, ths sends black/black crossngs and dots to the usual natural transformatons and a a 1 c d g h b 1 b. Proof. Of course, all relatons only nvolvng black strands are already confrmed by [Webb, 1.6]. Smlarly the 2-colored relatons of Fgure 3 are clear, snce ths ust the statement that a and a 1 are nverse (and smlarly for b ). Smlarly, the compatblty of the red/black or blue/black crossngs follows from the defnton of c, d, g, h from the unts and counts of these adunctons. The relatons of Fgure 7 follow from our descrpton of c and d (resp. g and h ) n terms of the surectve maps from R to the correspondng bmodules; n both cases one s nduced by the dentty map, and one by y, where as before, y denotes the dot endomorphsm of the functor F or E. Fnally, the red relatons and blue relatons of Fgure 6 correspond under applyng the vertcal reflecton nvoluton to all bmodules. Somewhat confusngly, ths swtches left and rght actons, reversng the order of applcaton of functors, and thus correspondng to a horzontal (around the y-axs) reflecton of 1-morphsms; ths accounts for the sgn change between the frst 2 lnes. Thus, we only check the red relatons. Each of these follows from a smple calculaton nvolvng the natural map from R. The lower two lnes follow from the fact that a s nduced by the dentty map on R. The frst red lne follows from the commutaton relaton of a power of a dot wth a 15

16 crossng: = + b a a+b+1= δ, Thus, we have shown all relatons, and the result follows. Lemma 2.2 We have the followng equaltes of nduced functors = = = d 1 2 = 2 1 f 1 = 0 for all. 2 Applyng these relatons nductvely, any sequence of ϑ s and ε s can be reduced one of the form ϑ ν µε µ. The morphsm space between any two sequences (,, κ) and (,, κ ) has an obvous spannng set analogous to that of Khovanov and Lauda, defned by a sngle mnmal dagram for each (, )-parng n the sense of [KL10, 2.2]. That s for each parng on the concatenaton of and whch matches elements from dfferent sequences wth the same sgn or the same sequence wth dfferent sgns, we choose a dagram whch wres up each par wth a mnmal number of crossngs, and fx a pont on each wre. Each bass vector corresponds to a choce of a (, )-parng, a non-negatve nteger for each strand, and a monomal n the negatvely orented bubbles (ncludng fake bubbles). We construct the bass vector by takng the chosen mnmal dagram for that parng, addng the number of dots attached to each strand, and multplyng on the far rght by the monomal n the bubbles. We denote ths set C. Of course, there s no choce for how to wre up the weghts whch appear; n partcular, smply deletng the red and blue lnes gves a becton wth Khovanov and Lauda s spannng set for Mor U (, ). ( Proposton 2.3 The set C s a bass for morphsms Mor T (,, κ), (,, κ ) ) 16

17 Ben Webster Proof. The proof that these are a spannng set s essentally equvalent to that of [KL10, 3.11]; any two mnmal dagrams for the same parng are equvalent modulo those wth fewer crossngs (usng the relatons). Smlarly, movng dots to the chosen postons only ntroduces dagrams wth fewer crossngs. Thus, we only need show that all mnmal dagrams span. Of course, f a dagram s non-mnmal then t can be rewrtten n terms of the relatons n terms of ones wth fewer crossngs. Thus, by nducton, ths process must termnate at a expresson n terms of mnmal dagrams. The proof that these vectors are a bass proceeds by showng that any non-trval lnear combnaton acts non-trvally on the sum V. So, assume not, and let X + z = c C a c c be a fnte sum of elements of C whch acts trvally on V, and let X + c be an element for whch a c 0 and the number of crossngs n c s maxmal among elements of these property. We let κ = 0 be the ( constant functon on [1, l], and consder the elements of the Hom-space Mor T (,, κ ), (,, κ ) ) obtaned multplyng wth θ κ and θ κ, the elements whch sweep black strands to the rght and red or blue to the left. Rewrtng θ κ z θ κ n terms of spannng set we have chosen for ths Hom-space, we see that the leadng order term of θ κ c θ κ s that correspondng to the same one of Khovanov and Lauda s bass vectors, and ths appears wth the same multplcty a c. Obvously, ths element must also act trvally on V, so t suffces to only consder elements X + where κ = κ = 0. Of course, by basc lnear algebra, t suffces to prove that the Mor-spaces have the correct graded dmenson (snce dmensons of all graded peces are fnte). Moreover, snce we have a U-acton, we can always wrte any obect as a summand of a drect sum of obects where all F s are appled before all E s, and so t suffces to prove the dmenson formula for these. Usng the baduncton between F and E, can then move all E s out of the doman, reorder, and then move all of them out of the target. Thus, t suffces to prove the dmenson formula for both doman and target whch only use F s. Thus consder a z as before n ( ths case. There s a obvous map R Λ Mor T (,, κ ), (,, κ ) ) where we dentfy the polynomals n postve bubbles wth the graded rng Λ of symmetrc polynomals n nfntely many varables, sendng clockwse orented bubbles to elementary symmetrc functons (and thus counter-clockwse bubbles to the complete symmetrc functons, up to sgn). Every spannng set element corresponds to a natural bass vector of R Λ, and we can use these to construct a lft of z whch we denote by z. Now, consder the functor formed by the collecton of red and blue lnes; by Lemma 2.2, ths s the same functor as ϑ ν µε µ for some weghts, µ, ν. By [Webb, 1.6], the mage of the element z gves a non-zero map F M F M for some obect M of V µ for some domnant weght. Our clam s that f we nstead act on ϑ µm by the functors F ϑ ν µε µ and F ϑ ν µε µ, then z wll nduce a non-trval map. Of course, t could not harm 17

18 matters to apply the functor ε ν µ. Then we note that ε ν µf ϑ ν µε µϑ µm = ε ν µf ϑ ν µm = F ε ν µϑ ν µm = F M and the acton of z s ust that nduced by z, and thus s not 0. Thus, we have arrved at a contradcton, and the result s proved. Just as on U, the 2-category T has an autofunctor flppng dagrams whch s covarant on 1-morphsms and contravarant on 2-morphsms, whch we wll abuse notaton and also denote ψ. 3. Tensor product algebras As we mentoned, the category T s qute auxlary from our perspectve. The fundamental obect of ths paper, rather s an nduced module category over ths 2-category. Recall that the category U has a trval representaton on Vect k. Every 1- morphsm correspondng to a non-empty sequence acts trvally on any obect, as does the dentty 1-morphsm of any non-zero weght, whle d 0 V V for all vector spaces. Defnton 3.1 We let X denote the nducton of ths representaton to T. That s, an obect of X s a sum of 1-morphsms of T formally appled to obects of Vect k. In addton to the morphsms gven by tensor products, we also add a natural somorphsm tu V t uv t Mor T (, µ), u Mor U (µ, ν), V Ob(Vect k ). Remember that our conventon for swtchng between formulas and dagrams s dyslexc; t swtches left and rght. In essence, thus X s the quotent of all dagrams n T (whch we vew as obects n X by tensorng wth k tself) wth a F or E or a weght other 0 at the far left, snce we can move these over to act (trvally) on the vector space k. The reader s free to magne the obect k as a horde of zombes at the far left of the plane whch hungrly eats any black strand or non-zero weght t can reach, but whch s unable to pass through red or blue lnes. The category X stll carres a U acton by horzontal composton on the rght, but s far from rreducble or ndecomposable, snce U s unable to change the labelng or orderng of the red and blue strands. Defnton 3.2 Let X denote the U-nvarant subcategory consstng of all 1-morphsms (now thought of as obect of X) where the sequence of labels on red and blue lnes s exactly. Let X µ be the subcategory of X where the weght at the far rght s µ. 18

19 Ben Webster Recall that the author has already defned a categorfcaton of the tensor product of hghest weght representatons, based on certan algebras T, defned n [Webb, 2]; these are, n fact, a specal case of the categorfcatons we dscuss n ths paper. Theorem 3.3 If conssts only of domnant weghts, then X T pmod. Proof. Obvously, f one takes the drect sum over all sequences usng negatve smple roots, then one fnds an obect whose endomorphsms are T. Thus, we need only show that such sequences generate our category (as an addtve, dempotent complete category). Usng the relatons of T to pass all E s leftward (n the dagram; toward the zombes), we can always wrte a sequence as an summand n a sum of sequences wth E s further left or wth fewer E s; here t s crucal that there are no blue strands, snce E s cannot pass through these. By nducton, every sequence s a summand of sequences where all E s come at the far left. Thus, the only ones that gve non-zero obects are those wth no E s at all. As usual, we let X = X (). Theorem 3.4 If g s fnte type, we have an equvalence of U-module categores X X w 0. Proof. By symmetry, t suffces to assume s domnant. If Y s a U-module category, and M Ob(Y ) s a obect klled by all E and all postve degree bubbles, then there s a obvous map T Mor Y ( F M), and thus a U-equvarant functor X Y sendng P M. Now, consder X w0 ; by the same argument as Theorem 3.3, ths category s generated by sequences begnnng wth the only blue lne (at the left) and then only usng E s. Obvously, the endomorphsms of the sum of all these obects s an algebra wth defnton analogous to T ; these algebras are n fact somorphc under the map that turns red strands blue, reverses all orentatons, and multples by 1 rased to the number of crossngs wth the same label on them. We could, of course, also apply a Dynkn dagram automorphsm. Thus, we have an equvalence X w 0 µ Tw 0 µ pmod, but ths s, of course, not an equvalence of U-module categores; t doesn t preserve weght. However, t does tell us that X w 0 Vect. In partcular, the unque smple module of ths weght must be klled by all F s (snce these go to empty weght spaces) and by all postve degree bubbles. Ths nduces a functor X X w0 whch s an equvalence on the hghest weght space. From the explct descrpton of the smple of the lowest weght space gven n [Webc], t also nduces an equvalence on the lowest weght space. Ths 19

20 k k w 0 µ w 0 µ Fgure 8. The argument for fullness n Proposton 3.5 shows that ths functor s essentally surectve and full, snce the every obect n X w 0 s a summand of a sum of E s appled to the lowest weght vector. We have already calculated that the Euler forms of these representatons are both are gven by the Shapovalov form; ths follows for X by [Webb, Corollary 1.7], and for X w0 by the untarty of the Shapovalov form. Thus, they concde, and the functor must also be fathful. Proposton 3.5 If g s fnte type, we have an equvalence of U-module categores X (,µ) X (w 0,µ). Proof. We may as well assume that and µ are domnant, snce all other cases follow from ths one by symmetry. The equvalence must send the sequence (, µ) to (w 0,, µ) where (w 0, ) gves the unque ndecomposable obect n the -weght space of X w0. Such a functor exsts snce (w 0,, µ, ) s klled by the th power of the dot the last E, and smlarly, (w 0,, µ, ) s klled by the µ th power of the dot on last F. Snce the ungraded Euler forms of the 2 categores concde by Theorem 3.10, we need only prove that ths functor s full. Now, consder a morphsm between (w 0,, µ, ) and (w 0,, µ, k), where and k are arbtrary sequences n U. We wsh to show that ths s nduced by a 2-morphsm n U from k. When we draw the dagram of such a morphsm s that a termnal n at the bottom mght connect to one n or k (correspondng to ether a E or a F respectvely). A dagram of the second typemust have a crossng between the strand passng from to k and one passng from to the copy of at the top; ths crossng can be pulled left untl t occurs left of the red lne for µ. Thus ths dagram factors through a sequence of the form (w 0,, α,... ) for some α ; but (w 0,, α ) 0 snce the -weght space of V Z w 0 s trval. Thus, ths dagram s 0, and by nducton, we can wrte our dagram wth no strands from connectng to k. Ths argument s represented schematcally n the frst pcture of Fgure 8. If there are any strands connectng to, then there must be at least one strand opposte t whch arcs from the top copy of to k. We can push these strands together 20

21 Ben Webster at the cost of correcton terms where fewer strands pass from to, and then move the left crossng of the bgon left untl we agan factor through (w 0,, α,... ), and we can thus use the argument from above to see that ths dagram s 0. Ths argument s represented schematcally n the second pcture of Fgure 8. Ths shows the fullness of the functor and completes the proof. Whle the equvalence we have gven X µ T µ pmod can easly be extended to an equvalence X µ X µ (even wth no fnte type assumpton), ths s, of course, not an equvalence of U-representatons. As n [Webb, 3.12], we defne a strngy sequence to be one where the sequence of roots between any two weghts s All negatve (postve) f the leftward weght s (ant-)domnant. The strng parametrzaton of a non-zero element n the crystal graph of V n terms of the lowerng (rasng) Kashwara operators f s (ant-)domnant. Now, we wsh to analyze more serously the structure of these categores. Snce we wll use ths fact many tmes, let us remnd the reader that n a graded category where the degree 0 part of the endomorphsms of any obect are fnte dmensonal (a condton satsfed by U, T and X ), an obect s ndecomposable f and only f ts endomorphsm algebra s graded local,.e. has a unque maxmal homogeneous deal. Lemma 3.6 Every ndecomposable obect of X s a summand of a strngy sequence. Proof. Frst, we clam that t s suffcent to show ths for k a feld. We let r = k/m be the resdue feld of k, and assume that the theorem holds n ths case. We have a natural functor X k X r gven by smply kllng m. Gven a 1-morphsm P n X k, we can consder ts reducton P. By assumpton ths s a summand of a strngy sequence (, κ) (we nclude the bar to ndcate we consder t n X r), so we have maps P (, κ) P whch compose to the dentty. By Hensel s lemma, these lft to maps P (, κ) P whose composton s an somorphsm (and thus may be assumed to be the dentty). Thus, P s a summand, and we have reduced to the case of a feld. Now, we nduct on l; for l = 1, ths follows from [KL09, 3.20]. Now, when we consder general l, we can assume by nducton that our chosen ndecomposable s summand of a sequence where the elements left of the last red or blue strand are a strngy sequence. Thus, we need only show that the elements to the rght of the last strand can also be taken to be a strng parametrzaton. Assume for smplcty that the last strand s red (we can pass to the blue case usng the automorphsm whch flps red and blue). Then we also nduct on the number of black strands labeled wth negatve roots after the last red. Frst, we apply the argument of 3.3 to show that we can remove all strands wth postve labels. Thus, we can thnk of ths sequence as comng from actng by a 1-morphsm P n T whch has 21

22 only one red lne labeled wth l and a sequence black lnes labeled wth negatve roots appled to the orgnal sequence truncated at the last red lne. In fact, we can replace by the mage of a prmtve dempotent n the KLR algebra R End U (). By [LV11, 7.4], any ndecomposable proectve R ν -module can be wrtten as a summand of a sequence correspondng to strng parametrzatons n the crystal B( ). Applyng the same dempotent, we can thus fnd our orgnal ndecomposable nsde a sequence whch s strngy up to the last red lne, and then followed by a strng parametrzaton n B( ). If ths crystal element n B( ) has non-zero mage n the crystal of V l, then our sequence s strngy and we are done. Otherwse, ths means that no quotent of the correspondng R-module s klled by the cyclotomc deal for l n R; thus P s a summand of a 1-morphsm n T whch stll only has one red lne and black only labeled wth negatve roots, but at least one black left of the red. Thus, we have wrtten t as a summand of a sequence wth fewer black lnes rght of the last red, and by nducton we are done. Defnton 3.7 We defne an orderng on compostons of length l called reverse domnance order by ν ν f and only f l k= ν l k k= ν k for all [1, l]. If ν = ν, then ths concdes wth the usual domnance order. Smlarly, we let reverse lexcographc order on sequences of ntegers be that where a a f the rghtmost entry where the sequences dsagree s larger n a. We order strngy sequences by reverse domnance order on the composton gven by the numbers of black strands between successve red and blue strands, wth reverse lexcographc order breakng tes (here we thnk of the strngy sequence as extended nfntely leftward wth 0 s). Proposton 3.8 Each strngy sequence I has at most one summand whch s not somorphc to any summand of a larger sequence, whch s absolutely ndecomposable; every ndecomposable occurs n ths poston for a unque strngy sequence. Proof. As n the proof of Proposton 3.6, we can mmedately reduce to the case where k s a feld usng Hensel s lemma. Let us nduct on length of our sequence, and the orderng we have gven. Let I be a strngy sequence; then ths sequence I wth ts last block of n removed s agan strngy, and thus, by nducton, the sum of one ndecomposable whch s new and other ndecomposables assocated to hgher sequences. That s, we may assume that there s a unque proper maxmal graded deal I of End(I ) whch contans all elements factorng through shorter sequences. Now consder the deal I n End(I) generated by the deal I 22

23 Ben Webster the unque maxmal deal of End( θ n n ) all maps factorng through hgher strngy sequences We wsh to show I s maxmal, n whch case there wll only be at most one summand t does not kll. For any d End(I), ths element can be rewrtten as a sum d = a + b of elements a wth no crossngs between the groups of consecutve lke colored strands n the strngy sequence plus one b of elements where the rghtmost crossng or cup/cap s at the left edge of a group of lke colored strands θ m m. If what happens at the far rght n b s a crossng, lookng at the sequence n the mddle of ths dagram where ths crossng happens, we have a group of θ m + 1 strands colored m and all strands to the rght agree wth the strngy sequence. If t s a cap, then above the cap, we have a sequence hgher n reverse domnance order. Thus, n ether case, the element b factors through hgher strngy sequences. On the other hand, a s ust a product of elements n End( θ m m ) for all m; the product of the mages n End(I) s a graded local subrng wth quotent k, so a = a + 1 for some a I. Thus d = d + 1 where d I; that s I s maxmal and the quotent by t s a feld (or trval), so there s at most one ndecomposable summand of I not klled by I; and ths summand s absolutely ndecomposable. Ths proof s easly extended to show that f, range over all postve strng parametrzatons for elements of the crystal B( ), then the 1-morphsms (, ), whch we wll also call strngy and endow wth the same order, satsfy a smlar property. Proposton 3.9 The 1-morphsm (, ) n U has at most one summand, whch s not a summand of any hgher strngy sequence. Ths summand s absolutely ndecomposable and every ndecomposable appears ths way for a unque strngy sequence. As n [Webb, 3.2], we can defne vectors v κ n V Z nductvely by f κ(l) = n, then v κ = v κ v l where v l s the hghest (lowest) weght vector of V l f l s domnant (ant-domnant), and κ s the restrcton to [1, l 1]. If κ(l) n, so v κ = E n v κ, where = ( 1,..., n 1 ), usng the conventon that F = E. Theorem 3.10 The Grothendeck group K 0 (X ) s somorphc to V Z, ntertwnng the Euler form wth the factorwse Shapovalov form, s on the tensor product. Proof. We clam that ( ) dm Mor(P κ, Pκ ) = v κ, v κ s. We prove ( ) by nducton on n and l. Unless n = κ(l) = κ (l), we can move a F from one sde to become a E on the other (up to shft). Ths has the same effect on the Shapovalov form, snce E and F are badont under, s. The decompostons of 23

24 E P κ nto Pκ s matches that of the vector snce both are done usng the commutaton relatons between E and F or E and F, whch we already know match. If n = κ(l) = κ (l), then the dmenson of the Mor-space and the nner product are both unchanged by smply removng the red lne. Ths shows the equalty ( ). Thus, f we are gven any lnear relaton satsfed by [P κ ] s, the correspondng lnear combnaton of v κ s s n the kernel of ths form, and thus 0 n V. Thus, [P κ] vκ defnes a surectve map. Thus, we need only show that ths map s nectve. By Lemma 3.8, the strngy sequences span K 0 (X ); on the other hand, the are clearly sent to a bass of V Z snce they are upper trangular wth any crystal bass. Of course, any lnear map sendng a spannng set to a bass s an somorphsm. Ths n partcular shows that the classes of strngy sequences are lnearly ndependent, so the new summand of each strngy sequence must be non-zero. Corollary 3.11 Each strngy sequence n X or U has exactly one ndecomposable non-zero summand whch s not somorphc to any summand of a larger sequence. The autofunctor ψ obvously preserves volatng morphsms, and thus descends to an nvoluton on X whch we denote ψ. Let and µ both be domnant weghts. Proposton 3.12 The 2-functor ψ categorfes the bar nvoluton of V. The 2-functor ψ,µ categorfes the nvoluton Ψ of V V µ as defned n [Lus92]. Proof. The nvoluton Ψ on V V µ s the unque nvoluton whch satsfes Ψ(u (v v µ )) = ū (v v µ ), so we need only show that ψ,µ satsfes ths property, whch s clear from ts defnton; ths s smply that flppng over a dagram commutes wth actng on (, µ) wth t. Ths functor defnes an nvoluton on V for any, whch has smlar propertes to bar nvolutons prevously defned (and agreeng wth them n all cases where they are defned). We wll denote ths nvoluton ψ. Proposton 3.13 For each ndecomposable proectve P n X (resp. U), there s a unque gradng shft P(n) such that ψ (P(n)) P(n) (resp. ψ(p(n)) P(n)) Proof. Such a shft s obvously unque, so we need only prove t exsts. There s a unque n such that P(n) s a summand of the correspondng strngy sequence. Snce the latter module s self-dual, ψ (P(n)) s a summand of t, and by the unqueness of Proposton 3.9, we must have ψ (P(n)) P(n). 24

25 Ben Webster 4. Representaton categores and standard modules As n [Webb, Webc], t wll be useful to deal wth an abelan category, not ust an addtve one. In partcular, (as far as the author s aware) ths s necessary to check that the Grothendeck group of X s the tensor product representaton as a representaton of U q (q); we have thus far only checked that ths holds at q = 1. Defnton 4.1 We let V be the category of representatons of X whch send any I to a fnte-dmensonal vector space, that s, the category of functors X gvect k. Let Y : X V be the Yoneda embeddng I Hom(I, ). Note that we do not requre an obect n V to be fntely generated. Defnton 4.2 Let H be the subrng of the endomorphsm rng End X klls all but fntely many summands. ( I I) whch Note that ths s a non-untal rng; by a H -module M, we always mean one whch s the drect sum of the mages of the dempotents e,,κ. We can nterpret an obect n V as a module over H usng the obvous functor X H pmod gven by the morphsm space X Mor X ( I, ). I Of course, V s an abelan category, and snce Y (P) s proectve for any P Ob(X ), V has enough proectves. However, t s not clear that f M s fntely generated, and P s a fntely generated proectve wth a surecton P M, then the kernel of ths map s fntely generated. We let V = D fg (V ) be the trangulated category of complexes of fntely generated proectves whch are bounded above. We can defne an acton of U on V by exact functors usng the baduncton between E and F as a defnton,.e. F M(I) := M ( E I( α, µ α ) ) E M(I) := M(F I). Theorem 4.3 The Grothendeck group K 0 (V ) s somorphc to the lattce dual to V Z, wth the map nduced by Y gven by the Shapovalov form. Snce we have twsted our acton by the Cartan nvoluton, the Shapovalov parng defnes a map of representatons. If g s nfnte-type, then we take the full dual; that s, as abstract abelan groups, V Z s a drect sum of copes of Z, whle K0 (V ) s a drect product. We let V = K 0 (V ) Z C; ths s a g representaton defned by takng drect product of the weght spaces of a hghest weght representaton, rather than drect sum. Even n fnte-type, the Shapovalov form s not always unmodular over the ntegers, so ths wll not usually concde wth K 0 (X ); ths wll only happen f all entres n are mnuscule and g s fnte-type. In partcular, ths shows that V extremely 25

26 rarely has fnte global dmenson, snce that s only possble when these lattces concde. As n [LV11] and [Webb, 3.3], we can place a crystal structure on the set B of somorphsm classes of smple obects of V. We let ẽ M := soc(e M) f M := cosoc(f M). Theorem 4.4 These operators make B nto a crystal somorphc to that of V. Proof. Ths s the same proof as [Webb, 3.9]. For a sequence I = (,, κ), the number of black strands between each par of non-black strands defne a composton, whch we denote by ν I. Defnton 4.5 The standard representaton S I s the maxmal quotent of Y (I) such that S I (I ) = 0 f ν I > ν I n the reverse domnance order on compostons. We thnk of the relaton nduced from reverse domnance order on compostons as a preorder of the set of sequences I. If we wrte I as the concatenaton of sequences I wth one red or blue strands and then black strands to ts rght, then we let s I = v I1 v Il Proposton 4.6 Under the somorphsm K 0 (X ) Z C V, we have that [S I ] = s I. Furthermore, the representaton E S I carres a fltraton by the standards for I () = I 1 I α I +1 I l shfted n degree by α, I 1 I 1 and the representaton F S I carres a fltraton by the standards for I ( ) = I 1 I α I +1 I l shfted n degree by α, I +1 I l Proof. The proof s exactly the same as [Webb, 3.7], so we only gve a sketch. The frst step s to construct the fltratons as the mage of the map from I (±) to I, shown n Fgure ± 1 +1 ± Fgure 9. The element x m nducng the fltraton on E ± S I ; the lnes shown as red could ust as easly be blue. 26