A diagrammatic categorification of the q-schur algebra

Transcription

1 Quantum Topol. 4 (23), 75 OI.47/QT/34 Quantum Topology European Mathematcal Socety A dagrammatc categorfcaton of the q-schur algebra Marco Mackaay, Marko Stošć and Pedro Vaz Abstract. In ths paper we categorfy the q-schur algebra S q.n; d/ as a quotent of Khovanov and Lauda s dagrammatc 2-category U.sl n / [6]. We also show that our 2-category contans Soergel s [33] monodal category of bmodules of type A, whch categorfes the Hecke algebra H q.d/, as a full sub-2-category f d n. For the latter result we use Elas and Khovanov s dagrammatc presentaton of Soergel s monodal category of type A; see[8]. Mathematcs Subect Classfcaton (2). 8R5; 4M5, 7B37, 6W99. Keywords. Categorfcaton, quantum groups, quantum gl n, q-schur algebra, Soergel category. Contents Introducton Hecke and q-schuralgebras The 2-categores U.gl n / and.n; d/ A 2-representaton of.n; d/ Comparsons wth U.sl n / The dagrammatc Soergel categores and.n; d/ Grothendeck algebras References Foremost, we want to thank Mkhal Khovanov for hs remark that we should be able to defne the Chuang Rouquer complex for a colored brad usng hs and Lauda s dagrammatc calculus [6]. That observaton formed the startng pont of ths paper. We also thank Mkhal Khovanov and Aaron Lauda for further nsprng conversatons and for lettng us copy ther L A TEX verson of the defnton of U.sl n / almost lterally. Secondly, we thank the anonymous referee for hs or her detaled suggestons and comments whch have helped us to mprove the readablty of the paper sgnfcantly (we hope). Ths work was fnanced by Portuguese funds va the FCT Fundação para a Cênca e Tecnologa through proect number PTC/MAT/53/28, New Geometry and Topology. Pedro Vaz was also fnancally supported by the Fundação para a Cênca e Tecnologa through the post-doctoral fellowshp SFRH/BP/46299/ 28. Marko Stošć was also partally supported by the Mnstry of Scence of Serba, proect 742.

2 2 M. Mackaay, M. Stošć andp.vaz. Introducton There s a well-known relaton, called Schur Weyl dualty or recprocty, between the polynomal representatons of homogeneous degree d of the general lnear group GL.n; Q/ and the fnte-dmensonal representatons of the symmetrc group on d letters S d. Recall that all rreducble polynomal representatons of GL.n; Q/ of homogeneous degree d occur n the decomposton of V d,wherev Q n s the natural representaton of GL.n; Q/. Instead of the GL.n; Q/-acton, we can consder the U.gl n /-acton, wthout loss of generalty. A key observaton for Schur Weyl dualty s that the permutaton acton of S d on V d commutes wth the acton of U.gl n /. Furthermore, we have QŒS d Š End U.gln /.V d / f n d. By defnton, the Schur algebra s the other centralzer algebra S.n;d/ End Sd.V d /: It s well known that both U.sl n / and U.gl n / map surectvely onto S.n;d/, for any d>. Therefore we can also defne S.n;d/ as the mage of the map U.gl n /! End Q.V d /; whch s the defnton used n ths paper. Both S.n;d/ and QŒS d are splt semsmple fnte-dmensonal algebras, and the double centralzer property above mples that the categores of fnte-dmensonal modules S.n;d/ mod and S d mod are equvalent, for n d. There are two more facts of nterest to us. The frst s that there actually exsts a concrete functor whch gves rse to the above mentoned equvalence. For n d, there exsts an embeddng of QŒS d n S.n;d/, whch nduces the so called Schur functor S.n;d/-mod! S d -mod: As t turns out, ths functor s an equvalence. The second fact of nterest to us s that the Schur algebras S.n;d/ for varous values of n and d are related. If n m, then S.n;d/ can be embedded nto S.m;d/. A more complcated relaton s the followng: for any k 2 N, there s a surecton S.n;d C nk/! S.n;d/: Ths surecton s compatble wth the proectons of U.gl n / and U.sl n / onto the Schur algebras. Wth these surectons, the Schur algebras form an nverse system. As t turns out, the proectons of U.sl n / onto the Schur algebras gve rse to an embeddng Mn U.sl n / d lm S.n;d C nk/: k

3 A dagrammatc categorfcaton of the q-schur algebra 3 To get a smlar embeddng for U.gl n /, one needs to consder generalzed Schur algebras. We do not gve the detals of ths generalzaton, because we wll not need t. We refer the nterested reader to [7]. All the facts recollected above have q-analogues, whch nvolve the quantum groups U q.gl n / and U q.sl n /, the Hecke algebra H q.d/,theq-schur algebra S q.n; d/, and ther respectve fnte-dmensonal representatons over Q.q/. If one s only nterested n the fnte-dmensonal representatons of U q.gl n / and U q.sl n /, whch can all be decomposed nto weght spaces, t s easer to work wth Lusztg s dempotented verson of these quantum groups, denoted PU.gl n / and PU.sl n /. In these dempotented versons, the Cartan subalgebras are replaced by algebras generated by orthogonal dempotents correspondng to the weghts. The kernel of the surecton PU.gl n /! S q.n; d/ s smply the deal generated by all dempotents correspondng to the gl n -weghts whch do not appear n the decomposton of V d. Thesamestrueforthekernelof PU.sl n /! S q.n; d/, usngsl n -weghts. We wll say more about PU.gl n / and PU.sl n / n the next secton. We are nterested n the categorfcaton of the q-algebras above, the relatons between them and the applcatons to low-dmensonal topology. By a categorfcaton of a q-algebra we mean a monodal category or a 2-category whose Grothendeck group, tensored by Q.q/, s somorphc to that q-algebra. As a matter of fact, all of them have been categorfed already, and some of them n more than one way. Soergel defned a category of bmodules over polynomal rngs n d varables, whch he proved to categorfy H q.d/. Elas and Khovanov gave a dagrammatc verson of the Soergel category. Gronowsk and Lusztg [2] were the frst to categorfy S q.n; d/, usng categores of perverse sheaves on products of partal flag varetes. Subsequently Mazorchuk and Stroppel constructed a categorfcaton usng representaton theoretc technques [28] and so dd Wllamson [39]forn d usng sngular Soergel bmodules. Khovanov and Lauda have provded a dagrammatc 2-category U.sl n / whch categorfes PU.sl n /. Rouquer [32] followed a more representaton theoretc approach to the categorfcaton of the quantum groups. The precse relaton of hs work wth Khovanov and Lauda s remans unclear. We note that the categorfcatons mentoned above have been obtaned for arbtrary root data. However, ths paper s only about type A and we wll not consder other types. Our nterest s n the dagrammatc approach, by whch H q.d/ and U q.sl n / have already been categorfed. The goal of ths paper s to defne a dagrammatc categorfcaton of S q.n; d/. Recall that the obects of U.sl n / are the weghts of sl n,whch label the regons n the dagrams whch consttute the 2-morphsms. Our dea s qute smple: defne a new 2-category U.gl n / ust as U.sl n /, but swtch to gl n -weghts, whch we conecture to gve a categorfcaton of PU.gl n /. Next we mod out U.gl n / by all dagrams whch have regons labeled by weghts not appearng n the decomposton of V d. Ths way we obtan a 2-category.n; d/ and the man result of ths paper s the proof that t ndeed categorfes S q.n; d/. There are two good reasons for swtchng to gl n -weghts, besdes gvng a con-

4 4 M. Mackaay, M. Stošć andp.vaz ectural categorfcaton of PU.gl n /. It s easer to say explctly whch gl n -weghts do not appear n V d, as we wll show n the next secton. Also, whle workng on our paper we found a sgn mstake n what Khovanov and Lauda call ther sgned categorfcaton of PU.sl n /;see[7]. Fortunately t does not affect ther unsgned verson, but the corrected sgned verson loses a nce property, the cyclcty. We dscovered that wth gl n -weghts there s a dfferent sgn conventon whch solves the problem, at least for.n; d/. On our way of provng the man result of ths paper we obtan some other nterestng results. For n d, we defne a fully fathful 2-functor from Soergel s category of bmodules to.n; d/, whch categorfes the well-known ncluson H q.d/ S q.n; d/ explaned n Secton 2. We defne functors.n; d/!.m; d/ when n m. We are not (yet) able to prove that these are fathful, although we strongly suspect that they are. We know that they are not full, but suspect that they are almost full n a sense that we wll explan n Secton 7. We defne essentally surectve full 2-functors.n; d C kn/!.n; d/ whch categorfy the surectons above. We show that Khovanov and Lauda s 2-representaton of U.sl n / on the equvarant cohomology of flag varetes descends to.n; d/. We conecture how to categorfy the rreducble representatons of S q.n; d/ usng.n; d/. Khovanov and Lauda s categorfcaton of these representatons, usng the so-called cyclotomc quotents, should be equvalent to a quotent of ours. Understandng the precse relaton wth the other categorfcatons of S q.n; d/ would be very mportant, but s left for the future. As a matter of fact, Brundan and Stroppel have already establshed a lnk between the category O approach to categorfcaton and Khovanov and Lauda s (see for example [2]), whch perhaps can be used to obtan an equvalence between Mazorchuk and Stroppel s categorfcaton of the q-schur algebra and ours. For n d, Wllamson s 2-category of Soergel s sngular bmodules s equvalent to Khovanov and Lauda s 2-category buld out of the equvarant cohomology of partal flag varetes (of flags n Q d ) and we expect both to be equvalent to.d; d/. Besdes the ntrnsc nterest of.n; d/, wth ts combnatorcs and ts lnk to representaton theory, there s also a potental applcaton to knot theory. Frst recall that there s a natural surecton of the brad group onto H q.d/. The Jones Ocneanu trace of the mage of a brad n H q.d/ s equal to the so called HOMFLYPT knot polynomal of the brad closure. Ths constructon has been categorfed: Rouquer defned a complex of Soergel bmodules for each brad and Khovanov dscovered

5 A dagrammatc categorfcaton of the q-schur algebra 5 that ts Hochschld homology categorfes the Jones Ocneanu trace, showng that n ths way one obtans a homology whch s somorphc to the Khovanov Rozansky HOMFLYPT-homology. Usng Elas and Khovanov s work, Elas and Krasner [9] worked out the dagrammatc verson of Rouquer s complex. Ther work stll remans to be extended to nclude the Hochschld homology. Besdes ths approach, whch s the one most drectly related to the results n ths paper, we should also menton a geometrc approach due to Webster and Wllamson n [37] and a representaton theoretc approach due to Mazorchuk and Stroppel [29]. More generally, there s a natural homomorphsm from the colored brad group, wth n strands colored by natural numbers whose sum s equal to d, tos q.n; d/. It s not as wdely advertsed as the non-colored verson, but one can easly obtan t from Lusztg s formulas n Secton 5.2. n [22] or from the second part of the paper by Murakam Ohtsuk Yamada [3]. One can also defne a colored verson of the Jones Ocneanu trace on S q.n; d/ to obtan the colored HOMFLYPT knot nvarant. Naturally the queston arses how to categorfy the colored HOMFLYPT knot polynomal. In [5] Chuang and Rouquer defned a colored verson of Rouquer s complex for a brad, usng a representaton theoretc approach. They proved nvarance under the second brad-lke Redemester move and conectured nvarance under the thrd move. In [25] we defned a complex of sngular Soergel bmodules, whch s equvalent to the Chuang Rouquer complex. We conectured that the Hochschld homology of such a complex categorfes the colored HOMFLYPT knot polynomal of the brad closure. We were only able to prove our conecture for the colors and 2, due to the complexty of the calculatons for general colors. Webster and Wllamson subsequently showed our conecture to be true, usng a generalzaton of ther geometrc approach [38]. Cauts, Kamntzer and Lcata [3] also studed the Chuang Rouquer complex from a geometrc pont of vew. By the above mentoned 2-representaton of.n; d/ nto sngular Soergel bmodules, t s natural to expect that one should be able to defne the Chuang Rouquer complex n.n; d/ such that ts 2-representaton gves exactly the complex of sngular Soergel bmodules whch we conectured. In a forthcomng paper we wll come back to ths. In the meanwhle, papers have appeared n whch the colored HOMFLYPT homology has been constructed usng matrx factorzatons; see [4], [4], [42], [43], and [44]. The outlne of ths paper s as follows. In Secton 2 we recall some results on the above mentoned q-algebras. Our choce has been hghly selectve n an attempt to prevent ths paper from becomng too long. We have only ncluded those results whch we categorfy or whch we need n order to categorfy. We hope that ths ntroducton makes up for what we left out. In Secton 3 we defne the 2-categores U.gl n / and.n; d/. As sad before, the frst one s ust a copy of Khovanov and Lauda s defnton of U.sl n /, but wth a dfferent set of weghts and a dfferent sgn conventon. The second one s a quotent of the frst one.

6 6 M. Mackaay, M. Stošć andp.vaz To understand some of the propertes of.n; d/, we frst defne ts 2-representaton n the 2-category of bmodules over polynomal rngs n Secton 4. Except for the dfferent sgn conventon, t s the factorzaton of the 2-representaton of [6] through.n; d/. The only new feature s our nterpretaton of ths 2-representaton n terms of the categorfed MOY-calculus, whch we developed n [25]. Secton 5 s devoted to comparng the structure of the 2-HOM spaces of U.sl n / to those of.n; d/. The latter ones reman a bt of a mystery to us and we can only prove ust enough about them for what we need n the rest of ths paper. In Secton 6 we defne a fully fathful embeddng of Soergel s categorfcaton of H q.d/ nto.n; d/. We have not yet attrbuted any notaton to Soergel s category n ths ntroducton, because there are actually two slghtly dfferent versons of t and we wll need both, one for d n and the other for d<n. In Secton 7 we prove that.n; d/ ndeed categorfes S q.n; d/. We also conecture how to categorfy the Weyl modules of S q.n; d/. 2. Hecke and q-schur algebras In ths secton we recollect some facts about the q-algebras mentoned n the ntroducton. For detals and proofs see [6] and [27] unless other references are mentoned. We work over the feld Q.q/, whereq s a formal parameter. 2.. The quantum general and specal lnear algebras. Let us frst recall the quantum general and specal lnear algebras. The gl n -weght lattce s somorphc to Z n. Let ".;:::;;:::;/ 2 Z n, wth beng on the -th coordnate, and " " C 2 Z n,for ;:::;n. We also defne the Eucldean nner product on Z n by." ;" / ı ;. efnton 2.. The quantum general lnear algebra U q.gl n / s the assocatve untal Q.q/-algebra generated by K ;K,for;:::;n,andE,for ;:::;n, subect to the relatons K K K K ; K K K K ; K KC K K C E E E E ı ; ; q q K E q." ; / E K ;

7 A dagrammatc categorfcaton of the q-schur algebra 7 E 2 E.q C q /E E E C E E2 ; E E E E ; f ; else: efnton 2.2. The quantum specal lnear algebra U q.sl n / U q.gl n / s the untal Q.q/-subalgebra generated by K K C and E, for ;:::;n. Recall that the U q.sl n /-weght lattce s somorphc to Z n. Suppose that V s a U q.gl n /-weght representaton wth weghts. ;:::; n / 2 Z n,.e. V Š M V and K acts as multplcaton by q on V. Then V s also a U q.sl n /-weght representaton wth weghts N. N ;:::; N n / 2 Z n such that N C for ;:::;n. Conversely, gven a U q.sl n /-weght representaton wth weghts. ;:::; n /, there s not a unque choce of U q.gl n /-acton on V. We can fx ths by choosng the acton of K K n. In terms of weghts, ths corresponds to the observaton that, for any d 2 Z the equatons and C () nx d (2) determne. ;:::; n / unquely, f there exsts a soluton to () and (2) at all. To fx notaton, we defne the map ' n;d W Z n! Z n [fgby ' n;d./ ; f () and (2) have a soluton, and put ' n;d./ otherwse. Recall that U q.gl n / and U q.sl n / are both Hopf algebras, whch mples that the tensor product of two of ther representatons s a representaton agan. Both U q.gl n / and U q.sl n / have plenty of non-weght representatons, but we are not nterested n them. Therefore we can restrct our attenton to the Belnson Lusztg MacPherson [] dempotented verson of these quantum groups, denoted PU.gl n / and PU.sl n / respectvely. To understand ther defnton, recall that K acts as q on the -weght space of any weght representaton. For each 2 Z n adon an dempotent to U q.gl n / and add the relatons ı ; ; E E ; K q :

8 8 M. Mackaay, M. Stošć andp.vaz efnton 2.3. The dempotented quantum general lnear algebra PU.gl n / s defned by PU.gl n / M U q.gl n / : ;2Z n For. ;:::; m m /, wth 2f;:::;n g and,defne E E :::E n n and defne ƒ 2 Z n to be the n-tuple such that E Cƒ E : Smlarly for U q.sl n /, adon an dempotent for each 2 Z n and add the relatons ı ; ; E N E ; K K C q : efnton 2.4. The dempotented quantum specal lnear algebra PU.sl n / s defned by PU.sl n / M U q.sl n / : ;2Z n Note that PU.gl n / and PU.sl n / are both non-untal algebras, because ther unts would have to be equal to the nfnte sum of all ther dempotents. Furthermore, the only U q.gl n / and U q.sl n /-representatons whch factor through PU.gl n / and PU.sl n /, respectvely, are the weght representatons. Fnally, note that there s no embeddng of PU.sl n / nto PU.gl n /, because there s no embeddng of the sl n -weghts nto the gl n -weghts The q-schur algebra. Let d 2 N and let V be the natural n-dmensonal representaton of U q.gl n /.efne and ƒ.n; d / n 2 N n W nx o d ƒ C.n; d/ f 2 ƒ.n; d /W d 2 n g: Recall that the weghts n V d are precsely the elements of ƒ.n; d /, and that the hghest weghts are the elements of ƒ C.n; d/. The hghest weghts correspond exactly to the rreducbles V that show up n the decomposton of V d. As explaned n the ntroducton, we can defne the q-schur algebra as follows.

9 A dagrammatc categorfcaton of the q-schur algebra 9 efnton 2.5. The q-schur algebra S q.n; d/ s the mage of the representaton n;d W U q.gl n /! End Q.V d /. For each 2 ƒ C.n; d/, theu q.gl n /-acton on V factors through the proecton n;d W U q.gl n /! S q.n; d/. Ths way we obtan all rreducble representatons of S q.n; d/. Note that ths also mples that all representatons of S q.n; d/ have a weght decomposton. As a matter of fact, t s well known that S q.n; d/ Š Y End Q.V /: 2ƒ C.n;d/ Therefore S q.n; d/ s a fnte-dmensonal splt sem-smple untal algebra and ts dmenson s equal to X n dm.v / 2 2 C d : d 2ƒ C.n;d/ Snce V d s a weght representaton, n;d gves rse to a homomorphsm PU.gl n /! S q.n; d/; for whch we use the same notaton. Ths map s stll surectve and oty and Gaqunto, n Theorem 2.4 of [7], showed that the kernel of n;d s equal to the deal generated by all dempotents such that 62 ƒ.n; d /. LetPS.n; d/ be the quotent of PU.gl n / by the kernel of n;d. Clearly we have PS.n; d/ Š S q.n; d/. By the above observatons, we see that PS.n; d/ has a Serre presentaton. As a matter of fact, by Corollary n [4], ths presentaton s smpler than that of PU.gl n /: one does not need to mpose the last two Serre relatons, nvolvng cubcal terms, because they are mpled by the other relatons and the fnte dmensonalty. Lemma 2.6. PS.n; d/ ssomorphctotheassocatveuntalq.q/-algebra generated by,for 2 ƒ.n; d /, and E,for ;:::;n, subect to the relatons ı ; ; X ; 2ƒ.n;d / E E ; X E E E E ı Œ N : 2ƒ.n;d / We use the conventon that X,f or s not contaned n ƒ.n; d /. Recall that Œa s the q-nteger.q a q a /=.q q /. We thank Raphaël Rouquer for pontng ths out to us and gvng us the reference.

10 M. Mackaay, M. Stošć andp.vaz Although there s no embeddng of PU.sl n / nto PU.gl n /, the proecton n;d W U q.gl n /! S q.n; d/ can be restrcted to U q.sl n / and s stll surectve. Ths gves rse to the surecton n;d W PU.sl n /! PS.n; d/; defned by n;d.e / E 'n;d./; (3) where ' n;d was defned below equatons () and (2). By conventon we put. As mentoned n the ntroducton, the q-schur algebras for varous values of n and d are related. Let m n and d be arbtrary. There s an obvous embeddng of the set of U q.gl n /-weghts nto the set of U q.gl m /-weghts, gven by. ;:::; n / 7!. ;:::; n ;;:::;/: For fxed d, ths gves an ncluson ƒ.n; d / ƒ.m; d /, whch we can use to defne n;m X 2 PS.m; d/: Note that n;m unless n m. 2ƒ.n;d / efnton 2.7. There s a well-defned homomorphsm gven by n;m W PS.n; d/! n;m PS.m; d/ n;m E 7! n;m E n;m and 7! n;m n;m : It s easy to see that ths s an somorphsm. efnton 2.8. Suppose d d C nk, for a certan k 2 N. Then we defne a homomorphsm d ;d W PS.n; d /! PS.n; d/ by 7!.k n / and E 7! E : It s easy to check that d ;d s well-defned and surectve. It s also easy to see that d ;d n;d n;d and that d ;d nduces a lnear somorphsm V! V.k n /;

11 A dagrammatc categorfcaton of the q-schur algebra whch ntertwnes the PS.n; d / and PS.n; d/ actons, f.k n / 2 ƒ C.n; d/. Of course V and V.k n / are somorphc as U q.sl n / representatons. Furthermore, note that for any d ;:::;n the set.s q.n; d C nk/; dcnk;d / k2n (4) forms an nverse system, so we can form the nverse lmt algebra lm S q.n; d C nk/: k The followng lemma s perhaps a bt surprsng. Lemma 2.9. The map P d an embeddng We also have Q k n;dcnk, wth d ;:::;n and k 2 N, gves U q.sl n / PU.sl n / Mn d Mn d lm S q.n; d C nk/: k lm S q.n; d C nk/: (5) k The reader should remember ths embeddng when readng Corollary 5.2. The results n ths paragraph were taken from []. We need to recall two more facts about q-schur algebras and ther representatons. The frst s that the rreducbles V,for 2 ƒ C.n; d/, can be constructed as subquotents of PS.n; d/, called Weyl modules. Let < denote the lexcographc order on ƒ.n; d /. Lemma 2.. For any 2 ƒ C.n; d/, we have V Š PS.n; d/ =Œ > : Here Œ > s the deal generated by all elements of the form x,forsome x 2 PS.n; d/ and >. Fnally, we recall a well-known ant-nvoluton on PS.n; d/, whch we wll need n ths paper. efnton 2.. We defne an algebra ant-nvoluton by W PS.n; d/! PS.n; d/ op. / ;

12 2 M. Mackaay, M. Stošć andp.vaz Note that up to a shft t,wehave. C E / q N E C ;. E C / q C N C E : E s E s2 :::E sm E sm q t 7! E sm E sm :::E s2 E s q tct : Our s the analogue of the one n [6] The Hecke algebra. Recall that H q.n/ s a q-deformaton of the group algebra of the symmetrc group on n letters. efnton 2.2. The Hecke algebra H q.n/ s the untal assocatve Q.q/-algebra generated by the elements T, ;:::;n, subect to the relatons T 2.q 2 /T C q 2 ; T T T T f >; T T C T T C T T C : Note that some people wrte q wherewewrteq 2 and use v q n ther presentaton of the Hecke algebra. It s also not uncommon to fnd t nstead of our q. For q we recover the presentaton of QŒS n n terms of the smple transpostons. For any element 2 S n we can defne T T :::T k, choosng a reduced expresson ::: k. The relatons above guarantee that all reduced expressons of gve the same element T.TheT,for 2 S n, form a lnear bass of H q.n/. There s a smple change of generators, whch s convenent for categorfcaton purposes. Wrte b q.t C /. Then the relatons above become b 2.q C q /b ; b b b b ; f >; b b C b C b C b C b b C C b : These generators are the smplest elements of the so called Kazhdan Lusztg bass. Although the change of generators s smple, the whole change of lnear bases s very complcated. As mentoned n the ntroducton, there s a q-verson of Schur Weyl dualty. There s a q-permutaton acton of H q.d/ on V d, whch s nduced by the R-matrx of U q.gl n / or U q.sl n / and commutes wth the actons of these quantum envelopng algebras. Wth respect to these actons, H q.d/ and PS.n; d/ have the double

13 A dagrammatc categorfcaton of the q-schur algebra 3 centralzer property. Furthermore, ther respectve categores of fnte-dmensonal representatons are equvalent. Suppose n d. We explctly recall the embeddng of H q.d/ nto PS.n; d/. Let d. d /. Note that the U q.gl n /-weght. d / gves the zero U q.sl n /-weght, for n d, and a fundamental U q.sl n /-weght for n>d. We defne the followng map n;d W H q.d/! d PS.n; d/ d by n;d.b / d E E d d E E d ; for ;:::;d. It s easy to check that n;d s well-defned. It turns out that n;d s actually an somorphsm, whch nduces the q-schur functor PS.n; d/-mod! H q.d/-mod; where mod denotes the category of fnte-dmensonal modules. Ths functor s an equvalence. Let us state explctly an easy mplcaton of ths equvalence, whch we need n the sequel. Lemma 2.3. Let < d n and let A be a untal assocatve Q.q/-algebra. Suppose that W PS.n; d/! A s a surecton of Q.q/-algebras such that s an embeddng. Then A Š PS.n; d/. B n;d W H q.d/! A Proof. Recall that PS.n; d/ Š Y End Q.q/.V /: 2ƒ C.n;d/ The fact that the q-schur functor s an equvalence means that the proecton of n;d.h q.d// onto End Q.q/.V / s non-zero, for any 2 ƒ C.n; d/. Snce all End Q.q/.V / are smple algebras, A has to be somorphc to the product Y End Q.q/.V /; 2ƒ for a certan subset ƒ ƒ C.n; d/. But B n;d s an embeddng, so ƒ ƒ C.n; d/ has to hold.

14 4 M. Mackaay, M. Stošć andp.vaz 3. The 2-categores U.gl n / and.n; d/ In ths secton we defne two 2-categores, U.gl n / and.n; d/, usng a graphcal calculus analogous to Khovanov and Lauda s n [6]. We thank Khovanov and Lauda for lettng us copy ther defnton of U!.sl n /. Takng ther defnton, we frst ntroduce a change of weghts to obtan U.gl n /. Then we dvde by an deal to obtan.n; d/. As remarked n the ntroducton, our sgns are slghtly dfferent from those n [6]. Khovanov and Lauda [7] corrected ther sgn conventon n U!.sl n /. As t turns out, the corrected U!.sl n / s no longer cyclc, whch makes workng wth that sgn conventon awkward. Fortunately Khovanov and Lauda s non-sgned verson, U.sl n /,s stll correct and cyclc and s somorphc to the corrected U!.sl n /;see[6] and [7]. However, the sgn conventon n U.sl n / s not so practcal for the 2-representaton nto bmodules, so we have decded to stck to our own sgn conventon n ths paper. To get from our sgns back to Khovanov and Lauda s (corrected) sgns n U!.sl n /, apply the 2-somorphsm whch s the dentty on all obects, -and2-morphsms except the left cups and caps, on whch t s gven by ; 7!./ CC ; and 7! ;./ C : (6) ; The varous parts of our defnton of U.gl n / and.n; d/ below have exactly the same order as the correspondng parts of Khovanov and Lauda s defnton of U!.sl n /, so the reader can compare them n detal. From now on we wll always wrte U.sl n /, nstead of U!.sl n /, for the corrected sgned categorfcaton of PU.sl n /. Snce we wll never work wth the unsgned verson, there should be no confuson. 3.. The 2-category U.gl n /. As already remarked n the ntroducton, the dea underlyng the defnton of U.gl n / s very smple: t s obtaned from U.sl n / by passng from sl n -weghts to gl n -weghts. From now on let n 2 N > be arbtrary but fxed and let I f;2;:::;n g. In thesequelweusesgned sequences. ;:::; m m /, for any m 2 N, 2f g and 2 I. The set of sgned sequences we denote SSeq. For. ;:::; m m / 2 SSeq we defne ƒ. / ƒ CC m. m / ƒ,where. / ƒ.;;:::;;;:::;/; such that the vector starts wth and ends wth k zeros. To understand these defntons, the reader should recall our defnton of E and ƒ below efnton 2.3. We also defne the symmetrc Z-valued blnear form on QŒI by 2,. C/ and,for >. Recall that N C. efnton 3.. U.gl n / s an addtve Q-lnear 2-category. The 2-category U.gl n / conssts of the followng data. Obects: 2 Z n.

15 A dagrammatc categorfcaton of the q-schur algebra 5 The hom-category U.gl n /.; / between two obects, s an addtve Q-lnear category consstng of the followng data. Obects 2 of U.gl n /.; /: a -morphsm n U.gl n / from to s a formal fnte drect sum of -morphsms E ftg E ftg E E m m ftg for any t 2 Z and sgned sequence 2 SSeq such that C ƒ and, 2 Z n. Morphsms of U.gl n /.; /: for -morphsms E ftg and E ft g n U.gl n /, the hom sets U.gl n /.E ftg; E ft g/ of U.gl n /.; / are graded Q-vector spaces gven by lnear combnatons of degree t t dagrams, modulo certan relatons, bult from compostes of the followng 2-morphsms. ) egree zero dentty 2-morphsms x for each -morphsm x n U.gl n /;the dentty 2-morphsms EC ftg and E ftg, for 2 I, are represented graphcally by EC ftg E ftg C ƒ and ƒ deg deg for any C ƒ 2 Z n and any ƒ 2 Z n, respectvely. More generally, for a sgned sequence. ; 2 2 ;::: m m /, the dentty E ftg 2-morphsm s represented as 2 m C ƒ ::: 2 m 2 We refer to obects of the category U.gl n /.; / as -morphsms of U.gl n /. Lkewse, the morphsms of U.gl n /.; / are called 2-morphsms n U.gl n /.

16 6 M. Mackaay, M. Stošć andp.vaz where the strand labeled k s orented up f k Cand orented down f k. We wll often place labels wth no sgn on the sde of a strand and omt the labels at the top and bottom. The sgns can be recovered from the orentatons on the strands. ) For each 2 Z n the 2-morphsms Notaton: ; ; ;; ;; 2-morphsm: C ƒ C ƒ egree: Notaton: ; ; ; ; 2-morphsm: egree: C N N C N N Badontness and cyclcty. ) Cƒ E C and E C are badont, up to gradng shfts: 8 C ƒ C ƒ ; ˆ< ˆ: C ƒ 8 ˆ< C ƒ ˆ: C ƒ ; ; C ƒ C ƒ C ƒ : (7) (8)

17 A dagrammatc categorfcaton of the q-schur algebra 7 ) C ƒ C ƒ C ƒ. (9) ) All 2-morphsms are cyclc wth respect to the above badont structure. 3 Ths s ensured by the relatons (9), and the relatons : () Note that we can take ether the frst or the last dagram above as the defnton of the up-sde-down crossng. We have chosen the last one above, because t s the one whch matches Khovanov and Lauda s sgns. The cyclc condton on 2-morphsms expressed by (9) and () ensures that dagrams related by sotopy represent the same 2-morphsm n U.gl n /. It wll be convenent to ntroduce degree zero 2-morphsms: ; () : (2) where the second equalty n () and (2) follow from (). Agan we have ndcated whch choce of twsts we use to defne the sdeways crossngs, whch s exactly the choce whch matches Khovanov and Lauda s sgn conventons. 3 See [2] and the references theren for the defnton of a cyclc 2-morphsm wth respect to a badont structure.

18 8 M. Mackaay, M. Stošć andp.vaz v) All dotted bubbles of negatve degree are zero. That s, 8 f m< N, ˆ< ˆ: m f m< N, m (3) for all m 2 Z C, where a dot carryng a label m denotes the m-fold terated vertcal composte of or dependng on the orentaton. A dotted ; ; bubble of degree zero equals : 8./ C for N, ˆ< N (4)./ ˆ: C for N. ˆ< ˆ: N v) For the followng relatons we employ the conventon that all summatons are ncreasng, so that a summaton of the form P m f s zero f m<. 8 P N f N ; f P N g N Cg N Cf N g ; (5) N X fx f g N Cg f g N f ; (6)

19 A dagrammatc categorfcaton of the q-schur algebra 9 N X f fx g f g N f N Cg ; (7) for all 2 Z n (see () and (2) for the defnton of sdeways crossngs). Notce that for some values of the dotted bubbles appearng above have negatve labels. A composte of or wth tself a negatve number ; ; of tmes does not make sense. These dotted bubbles wth negatve labels, called fake bubbles, are formal symbols nductvely defned by the equaton C N N C and the addtonal condton./ C t CC N and N Cr CC t r C ::: N Cr./ C t r C ::: f N. (8) Although the labels are negatve for fake bubbles, one can check that the overall degree of each fake bubble s stll postve, so that these fake bubbles do not volate the postvty of dotted bubble axom. The above equaton, called the nfnte Grassmannan relaton, remans vald even n hgh degree when most of the bubbles nvolved are not fake bubbles. See [2]formore detals. v) NlHecke relatons: ; ; (9) We wll also nclude ()for as an sl 2 -relaton. : (2)

20 2 M. Mackaay, M. Stošć andp.vaz For ; : (2) The analogue of the R./-relatons. ) For 8 ˆ< ˆ:. A f, f. (22) Notce that. / s ust a sgn, whch takes nto account the standard orentaton of the ynkn dagram. ; : (23) ) Unless k and : (24) k k For. / : (25) The addtve Z-lnear composton functor U.gl n /.; / U.gl n /. ; /! U.gl n /.; / s gven on -morphsms of U.gl n / by E ft ge ftg 7! E ft C t g (26)

21 A dagrammatc categorfcaton of the q-schur algebra 2 for ƒ, and on 2-morphsms of U.gl n / by uxtaposton of dagrams C B C A 7! : Ths concludes the defnton of U.gl n /. In the next subsecton we wll show some further relatons, whch are easy consequences of the ones above Further relatons n U.gl n /. The followng U.gl n /-relatons result from the relatons n efnton 3. and are gong to be used n the sequel. Bubble sldes: N Cm 8 ˆ< ˆ: P m f.f m / C ƒ.c ƒ / Cm C ƒ.c ƒ / Cf mf f, f, (27) C N Cm C. C / ƒ.c.c/ ƒ / 2Cm C C. C / ƒ.c.c/ ƒ / Cm C ; (28)

22 22 M. Mackaay, M. Stošć andp.vaz N Cm C X f Cgm f C. C / ƒ..c/ ƒ / Cg ; C N Cm X f Cgm C. C / ƒ.c.c/ ƒ / Cg f ; (29) C N Cm C C. C / ƒ..c/ ƒ / 2Cm C. C / ƒ..c/ ƒ / Cm If we swtch labels and C, then the r.h.s. of the above equatons gets a mnus sgn. Bubble sldes wth the vertcal strand orented downwards can easly be obtaned from the ones above by rotatng the dagrams 8 degrees. More Redemester 3 lke relatons. Unless k we have and when k we have X k N 3Cf 4 f f 3 f 2 k C X g 2 g g 3 : N Cg 4 where the frst sum s over all f ;f 2 ;f 3 ;f 4 wth f C f 2 C f 3 C f 4 N and the second sum s over all g ;g 2 ;g 3 ;g 4 wth g C g 2 C g 3 C g 4 N 2. Note that the frst summaton s zero f N <and the second s zero when N <2. Redemester 3 lke relatons for all other orentatons are determned from (24), and (25), and the above relatons usng dualty. (3) (3)

23 A dagrammatc categorfcaton of the q-schur algebra Enrched Hom spaces. For any shft t, there are 2-morphsms W E C ftg H) E C ft 2g; W E C E C ftg H) E C E C ft g; W ftg H) E E C ft. C N /g; W E E C ftg H) ft. N /g; n U.gl n /, and the dagrammatc relaton gves rse to relatons n U.gl n / E ftg; E ft C 3 g for all t 2 Z. Note that for two -morphsms x and y n U.gl n / the 2hom-space Hom U.gln /.x; y/ only contans 2-morphsms of degree zero and s therefore fnte-dmensonal. Followng Khovanov and Lauda we ntroduce the graded 2hom-space HOM U.gln /.x; y/ M t2z Hom U.gln /.xftg;y/; whch s nfnte-dmensonal. We also defne the 2-category U.gl n / whch has the same obects and -morphsms as U.gl n /,butfortwo-morphsms x and y the vector space of 2-morphsms s defned by U.gl n /.x; y/ HOM U.gln /.x; y/: (32) 3.2. The 2-category.n; d/. Fx d 2 N >. As explaned n Secton 2,theq-Schur algebra PS.n; d/ can be seen as a quotent of PU.gl n / by the deal generated by all dempotents correspondng to the weghts that do not belong to ƒ.n; d /. It s then natural to defne the 2-category.n; d/ as a quotent of U.gl n / as follows. efnton 3.2. The 2-category.n; d/ s the quotent of U.gl n / by the deal generated by all 2-morphsms contanng a regon wth a label not n ƒ.n; d /.

24 24 M. Mackaay, M. Stošć andp.vaz We remark that we only put real bubbles, whose nteror has a label outsde ƒ.n; d /, equal to zero. To see what happens to a fake bubble, one frst has to wrte t n terms of real bubbles wth the opposte orentaton usng the nfnte Grassmannan relaton (8). In ths secton we defne a 2-functor 4. A 2-representaton of.n; d/ F Bm W.n; d/! Bm ; where Bm s the graded 2-category of bmodules over polynomal rngs wth ratonal coeffcents. Recall that n the prevous secton formula (32) we have defned the verson of a graded 2-category, as the 2-category wth the same obects and -morphsms, whle the 2-morphsms between two -morphsms can have arbtrary degree. In [6] Khovanov and Lauda defned a 2-functor G d from U.sl n/ to a 2-category equvalent to a sub-2-category of Bm. As one can easly verfy, G klls any dagram d wth labels outsde ƒ.n; d /, so t descends to.n; d/. In ths secton we have rewrtten ths 2-functor, whch we denote F Bm, n terms of categorfed MOY-dagrams, because we thnk t mght help some people to understand ts defnton more easly. For further comments see Secton Categorfed MOY dagrams. Before proceedng wth the defnton of F Bm, we frst specfy our notaton for MOY dagrams and ther categorfcaton. A colored MOY dagram [3], s an orented trvalent graph whose edges are labeled by natural numbers (ths label s also called the color or the thckness of the correspondng edge). At each trvalent vertex we have at least one ncomng and one outgong edge, and we requre that at each vertex the sum of the labels of the ncomng edges s equal to the sum of the labels of the outgong edges. Moreover, n ths paper we assume that all edges n MOY dagrams are orented upwards. To obtan a bmodule correspondng to a gven colored MOY dagram, we proceed n the followng way: To each edge labeled a, we assocate a varables, say x.x ;:::;x a /, and to dfferent edges we assocate dfferent varables. At every vertex (lke the ones n Fgure ), we mpose the relatons e.z ;:::;z acb / e.x ;:::;x a ;y ;:::;y b /; e.z ;:::;z acb / e.x ;:::;x a ;y ;:::;y b /; for all 2f;:::;aC bg, wheree s the -th elementary symmetrc polynomal. In other words, at every vertex we requre that an arbtrary symmetrc polynomal n

25 A dagrammatc categorfcaton of the q-schur algebra 25 the varables correspondng to the ncomng edges, s equal to the same symmetrc polynomal n the varables correspondng to the outgong edges. Now, to an arbtrary dagram, we assocate the rng R of polynomals over Q whch are symmetrc n the varables on each strand separately, modded out by the relatons correspondng to all trvalent vertces. In partcular, to a graph wthout trvalent vertces (ust strands) c b a ::: z y x we assocate the rng of partally symmetrc polynomals QŒx;y;:::;z S as b S c. In ths way, the rng R assocated to a MOY dagram, s a bmodule over the rngs of partally symmetrc polynomals assocated to the top (rght acton) and bottom end (left acton) strands, respectvely (remember that we are assumng that all MOY dagrams are orented upwards, so they have a top and a bottom end). Bmodules are graded by settng the degree of any varable equal to 2. In the rest of the paper, we wll often dentfy the MOY dagram and the correspondng bmodule. Also, by abuse of notaton, we shall call the elements of the bmodule R polynomals. There s another way to descrbe these bmodules assocated to MOY dagrams; see e.g. [3], [25], and [39]. Fx the polynomal rng R QŒx ;:::;x d. For any.a ;:::;a n / 2 ƒ.n; d /, letr a ;:::;a n be the sub-rng of polynomals whch are nvarant under S a S an. To the frst dagram n Fgure one assocates the R acb R a;b -bmodule Res RaCb R a;b R a;b ; where one smply restrcts the left acton on R a;b to R acb R a;b. To the second dagram n Fgure one assocates the R a;b R acb -bmodule Ind Ra;b R acb R acb R a;b RaCb R acb : In ths way, to every MOY-dagram one assocates a tensor product of bmodules, whch s somorphc to the bmodule R that we descrbed n the paragraph above. In ths paper we always use R, snce t s computatonally easer to use polynomals than to use tensor products of polynomals.

26 26 M. Mackaay, M. Stošć andp.vaz x ;:::;xa y ;:::;y b z ;:::;z acb a b a C b a C b a b z ;:::;z acb x ;:::;x a y ;:::;y b Fgure. trvalent vertces 4.2. efnton of F Bm. Now we can proceed wth the defnton of F Bm W.n; d/! Bm : Let z ;:::;z d be varables. For convenence we shall use Khovanov and Lauda s notaton k CC,for ;:::;n. On obects 2 ƒ.n; d /, the 2-functor F Bm s gven by. ;:::; n / 7! QŒz ;:::;z d S S n : On -morphsms we defne F Bm as follows: ftg 7! QŒz ;:::;z d S S n ftg: In terms of MOY dagrams ths s presented by: n 2 ftg 7! ::: Note that we are drawng the entres of from rght to left, whch s compatble wth Khovanov and Lauda s conventon. The remanng generatng -morphsms are mapped as follows: n C E C ftg 7! ::: ::: ft C C k C k k C g; C C n C E ftg 7! ::: ::: ft C k g: C C

27 A dagrammatc categorfcaton of the q-schur algebra 27 In both cases, the partton correspondng to the bottom strands s C ƒ (wth beng C or ). Thus, the condton we mposed on.n; d/ that all regons have labels from ƒ.n; d / (.e. no regon can have labels wth negatve entres), ensures that on the RHS above we really have MOY dagrams. The composte F Bm.E Cƒ E / s gven by stackng the MOY dagram correspondng to E on top of the one correspondng to E Cƒ. The shfts add under composton. To defne F Bm on 2-morphsms, we gve the mage of the generatng 2-morphsms. In the defntons the dvded dfference xy s used. For p 2 QŒx;y;:::t s gven xy p p p x$y ; (33) x y where p x$y s the polynomal obtaned from p by swappng the varables x and y. Moreover, for x.x ;:::;x a /, we use the shorthand x y@ x2 y :::@ xa y; yx2 :::@ yxa : (35) Before lstng the defnton of F Bm, we explan the notaton we are usng. We denote a bmodule map as a par, the frst term showng the correspondng MOY dagrams (of the source and target -morphsm), and the second beng an explct formula of the map n terms of the (classes of) polynomals that are the elements of the correspondng rngs. In a few cases we have added an ntermedate MOY-dagram, n order to clarfy the defnton. Fnally, n order to smplfy the pctures, n each formula we only draw the strands that are affected, whle on the others we ust set the dentty. Also n every lne we requre that the polynomal rngs correspondng to the top (respectvely bottom) end strands are the same throughout the move. Furthermore, we only wrte explctly the varables of the strands that are relevant n the defnton of the correspondng bmodule map: B 7! C C A ; r C B x #! C ;p7! x r C pa ;

28 28 M. Mackaay, M. Stošć andp.vaz C B 7! C A ; C C r C B B B B C B B B B 7! C C C C C x # x 2 # " x x # " x 2 x # " y x # " y!!!!! C C C C C2 C y C2 C # x # C2 C x C2 # C y # C!!!! 2 2 y # ;p7! x r C pa ;!! C C x # " x 2 x 2 # " x ;p7! p x 7!x 2 A ; " x 2 y # ;p7! p x7!x2 " x 2 C2 C x C2 # C y # C2 C y C2 C # x # C x x 2 pa ; C x x 2 pa ; C C A ; C ;p7! pa ; C ;p7!.x y/pa ; C ;p7! pa ; C ;p7!.x y/pa ;

29 C C A dagrammatc categorfcaton of the q-schur algebra 29 x C2 y # C C2 C # B y x 2 #! # ; p 7! p x 7!x 2 A ; (36) 7! C C B B B C2 C x # y # C2 x C # y # C2 C x # y #!!! C2 y # C # C2 C y x 2 # # C2 y # C # x 2 ;p7! px 7!x 2 C ;p7! p x 7!x 2 A ; C A ; (37) C A I x 2 ;p7! px 7!x 2 for 2 we have B 7! C C C A ; B 7! C C C A I sdeways crossngs for 2are defned n the same way as n the case of : x C C #! 7!. t ; P C p 7!./`x A `e`.t/p 7! p 7! ` C C %! z P C ` " y ; C A./`e C `.z/y`p

30 3 M. Mackaay, M. Stošć andp.vaz y C # C 7! u!! B " x A ; p ux.p yx / y C C # 7!! u B " x A : p xu.p yx / Ths ends the defnton of F Bm. Wthout gvng any detals, we remark that the bmodule maps above can be obtaned as compostes of elementary ones, called zp, unzp, assocatvty, dgon creaton and annhlaton, whch can be found n [25] F Bm s a 2-functor. We are now able to explan the relaton between our F Bm and Khovanov and Lauda s (see Subsecton 6.3 n [6]) G d W U.sl n/! EqFLAG d Bm : In the frst place, we categorfy the homomorphsm n;d from Secton 2. Note that all the relatons n.n; d/ only depend on sl n -weghts, except the value of the degree zero bubbles, whch truly depend on gl n -weghts. efnton 4.. We defne a 2-functor n;d W U.sl n /!.n; d/: On obects and -morphsms n;d s defned ust as n;d W PU.sl n /! PS.n; d/ n (3). On 2-morphsms we defne n;d as follows. Let be a strng dagram representng a 2-morphsm n U.sl n / (from now on we wll smply say that s a dagram n U.sl n /). Then n;d maps to the same dagram, multpled by a power of dependng on the left cups and caps n accordng to the rule n (6). The labels n Z n of the regons of are mapped by ' n;d to labels n Z n of the correspondng regons of n;d./,orto. Thsmeansthat,f has a regon labeled by such that ' n;d./ 62 ƒ.n; d /, then n;d./ by defnton. Fnally, extend ths defnton to all 2-morphsms by lnearty. It s easy to see that n;d s well-defned, full and essentally surectve. In the second place, recall that there s a well-known somorphsm QŒx ;:::;x d S S n Š H GL.d/.F l.k//; wth k.k ;k ;k 2 ;k 3 ;:::;k n /.; ; C 2 ; C 2 C 3 ;:::;d/,for any 2 ƒ.n; d / (see (6.25) n [6], for example). Usng ths somorphsm, t s

31 A dagrammatc categorfcaton of the q-schur algebra 3 straghtforward to check that the followng lemma holds by comparng the mages of the generators. Recall that G klls all dagrams wth labels outsde ƒ.n; d /. d Lemma 4.2. The followng trangle s commutatve: U.sl n / d G Bm : n;d F Bm.n; d/ The followng result s now an mmedate consequence of Khovanov and Lauda s Theorem 6.3. Proposton 4.3. F Bm defnes a 2-functor from.n; d/ to Bm. One could of course prove Proposton 4.3 by hand. We wll ust gve two sample calculatons. The result of the second one, the mage of the dotted bubbles, wll be needed n a later secton Examples of the drect proof of Proposton 4.3. We shall gve the proof for the zg-zag relaton of badontness and compute the mages of the bubbles by F Bm. Before proceedng, we gve some useful relatons that are used n the computatons. Frst of all, both the kernel and the mage of the dvded dfference xy consst of the polynomals that are symmetrc n the varables x and y. Ifp s symmetrc n the varables x and y xy.p q/ p@ xy q for any polynomal q. Also, note xy. We shall frequently use the followng useful denttes (see for example [] for the proofs). For x.x ;:::;x k /,leth.x/ denote the -th complete symmetrc polynomal n the varables x ;:::;x k. Then we have yx.y N / h N k.y; x/; (38) kx./ e.x/h k.x/ ı k; : (39) Moreover, f x, u.u ;:::;u a / and t.t ;:::;t ac / are varables such that e l.x; u/ e l.t/; l ;:::;ac ;

32 32 M. Mackaay, M. Stošć andp.vaz then for every l ;:::;ac we have lx e l.u/./ x e l.t/; (4) and e l.t/ e l.u/ C xe l.u/: (4) The zg-zag relatons. In order to reduce the number of subndces (to keep the notaton as concse as possble), we denote a and C b. Then the left hand sde of the frst of the relatons (7) s mapped by F Bm as follows: 7! b u & " x a t.! b u & x 2 # y " x a v t! b " x2 ap p x v./`x2 a` e`.v/p Note that p p.x ;u;t/ s symmetrc n the varables u and t separately. Also, the lowest trvalent vertex on the rght strand n the mddle pcture of the move, mples that e l.x ;v/ e l.t/, for every l ;:::;ac. So, x for > a s a symmetrc polynomal n the varables t (e.g. ths follows from (4) forl a C ). Thus we can wrte p as ax p x q.u;t/; (42) where q q.u;t/, ;:::;a, are polynomals symmetrc n u and t separately. Then we have ` a : C x v ax./ l x2 al e l.v/p l.l7!al/ ax./ l x2 x v e l.v/ l ax l ax x q ax./ al x2 l x v.x e al.v//: (43) Snce e l.x ;v/ e l.t/, for every l ;:::;ac, by(4) wehave Xal e al.v/./ k x k e alk.t/: k

33 A dagrammatc categorfcaton of the q-schur algebra 33 After replacng ths n (43), we x v ax./ l x2 al e l.v/p l (38) ax l ax l ax.k7!alk/ l ax x2 l q Xal./ alk e alk.t/@ x v.x Ck k / ax x2 l q Xal./ alk e alk.t/h Cka.x ;v/ k ax x2 l q Xal./ alk e alk.t/h Cka.t/ ax l k ax x2 l q Xal./ k e k.t/h lk.t/: Snce h p.t/ for p<, we must have k l. a l/ n the nnermost summaton,andsoby(39) the last expresson above s equal to ax l k ax x2 l q X l./ k e k.t/h lk.t/ k whch s ust the dentty map, as wanted. ax l ax x2 l q ı l; ax x 2 q p x 7!x 2 ; Images of bubbles by F Bm. Agan we denote a and C b. The clockwse orented bubble wth r dots on t s mapped by F Bm as follows: b a b a b a y % v! t.! % - 7! x t u. u : r bp C A p xv./`e b`.t/x`cr p ` The polynomal p p.t;u/ s symmetrc n the varables t and u separately. In partcular, we xv.p q/ p@ xv.q/, for any polynomal q. xv b X l./ l e bl.t/x lcr p bx./ l e bl.t/p@ xv.x lcr / l (38) p bx./ l e bl.t/h lcrac.u/ l

34 34 M. Mackaay, M. Stošć andp.vaz.l7!bl/ bx p./ b./ l e l.t/h bacrcl.u/: Snce e l.t/ for l>b,andh bacrcl.u/,forl>bacrc,wehave that the clockwse orented bubble s mapped by F Bm to the bmodule map bacrc X p 7! p./ b./ l e l.t/h bacrcl.u/: (44) l In partcular, f b a C r C <,.e. f r<ab, the bubble s mapped to zero, and f r a b, the bubble s mapped to./ b tmes the dentty (note that a b C s sl n weght). Also, r can be naturally extended to r a b (n (44)),.e. to nclude fake bubbles n the case a b. The counter-clockwse orented bubble wth r dots on t s mapped by F Bm as follows: r 7! b % t a - u! b v & t & y " x l a - u! ap p vx./`x a`cr e`.u/p ` Completely analogously as above, we have that the counter-clockwse orented bubble s mapped by F Bm to the bmodule map abcrc X p 7! p./ bc./ l e l.u/h abcrcl.t/: (45) l Agan, from r<b a, the bubble s mapped to zero, and f r b a, t s mapped to./ bc tmes the dentty. Moreover, r can be naturally extended to r b a,.e. to nclude fake bubbles n the case b a. Remark 4.4. Our reason for changng the sgns from [6], was to make the sgns n the mage of the degree zero bubbles,.e../ b for the clockwse bubble and./ bc for the counter-clockwse bubble, concde wth those of (4). Fnally, by the Gambell and the dual Gambell formulas (see e.g. []), from (44) and (45) the nfnte Grassmannan relaton follows drectly. b a : C A 5. Comparsons wth U.sl n / In ths secton we show the analogues for.n; d/ of some of Khovanov and Lauda s results on the structure of U.sl n /. Our results are far from complete. More work wll need to be done to understand the structure of.n; d/ better.

35 A dagrammatc categorfcaton of the q-schur algebra 35 To smplfy termnology, by a 2-functor we wll always mean an addtve Q-lnear degree preservng 2-functor. 5.. Categorcal nclusons and proectons. In the frst place, we categorfy the homomorphsms d ;d from Secton 2. efnton 5.. Let d d C kn, wth k 2 N. We defne a 2-functor d ;d W.n; d /!.n; d/: On obects and -morphsms d ;d s defned as d ;d. On 2-morphsms d ;d s defned as follows. For any dagram n.n; d / wth regons labeled 2 ƒ.n; d / such that.k n / 2 ƒ.n; d /,let d ;d./ be gven by the same dagram wth labels of the form.k n /, multpled by./ k for every left cap and left cup n. For any other dagram, let d ;d./. Extend ths defnton to all 2-morphsms by lnearty. Note that d ;d s well-defned, because N.k n /. The extra./ k for left cups and caps s necessary to match our normalzaton of the degree zero bubbles. It also ensures that we have d ;d n;d n;d ; where n;d W U.sl n /!.n; d/ s the 2-functor defned n efnton 4.. Note also that the d ;d form somethng lke an nverse system of 2-functors between 2-categores, because d ;d d ;d d ;d (compare to (4)). We say somethng lke an nverse system, because we have not been able to fnd a precse defnton of such a structure n the lterature on n-categores. Also one would have to thnk carefully f the nverse lmt of the.n; d/ would stll be Krull Schmdt. Fnally, there appears to be no general theorem that says that the Grothendeck group of an nverse lmt s the nverse lmt of the Grothendeck groups (even for algebras there s no such theorem). So we cannot (yet) reasonably conecture the categorfcaton of the embeddng (5). All we can say at the moment s the followng corollary. Corollary 5.2. We have () Let f be a 2-morphsm n U.sl n /. Let d >be the mnmum value such that Nˇ wth ˇ 2 ƒ.n; d /. Then f f and only f n;d Cnk.f / for any k. (2) Let ff g s be a fnte set of 2-morphsms n Hom U.sl n /.x; y/. Then the f are lnearly ndependent f and only f there exsts a d>such that the n;d.f / are lnearly ndependent n Hom.n;d/.x; y/.