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# Sem Analysis MATH 607

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4 THE REPRESENTATION THEORY OF THE SYMMETRIC GROUP In this chapter we give an overview of the p modular representation theory of the symmetric group Sn and its connection to the af ne Kac Moody algebra of type 141171 We work over a eld F of arbitrary characteristic p 41 Formal characters For k 1n we de ne the Jucys Murphy element kil mk Zn k 6 F5 1 i1 see 15 28 It is straightforward to show that the elements m1 m2 mn commute with one another Moreover we have by 15 or 29 19 Theorem 41 The center of the group algebra F5 is precisely the set of all symmetric polynomials in the elements 1 2 an Now let M be an FSn module Let I ZpZ identi ed with the prime sub eld of F For i i1 in E I de ne v M7i7Nv0forNgt0andeachr1n Thus M is the simultaneous generalized eigenspace for the commuting operators m1 zn corresponding to the eigenvalues i1 in respectively Lemma 42 Any FSn module M decomposes as M 161 Proof lt suf ces to show that all eigenvalues of x on M lie in I for each r 1n This is obvious if r 1 as 1 0 Now assume that all eigenvalues of x on M lie in I and consider 11 Let U E M be a simultaneous eigenvector for the commuting operators z and 711 Consider the subspace N spanned by v and 371 Suppose that N is two dimensional Then the matrix for the action of x on N with respect to the basis 125712 is E and c E F by assumption on the eigenvalues of 7 Hence the matrix for g for some ij E I i i 1 i the action of 711 srmrsr 37 on N is Z1 Z Since 1 was an eigenvector for 11 we see that c 71 hence v has eigenvalue j for 11 as required Finally suppose that N is one dimensional Then 31 iv Hence if my iv for i E I then 7111 srmrsr sv i1v Since i i 1 E I we are done El We de ne the formal character ch M of a nite dimensional FSn module M to be ch M Z dimMi ei 2 ZEIquot 1 2 an element of the free Z module on basis e1 E In This is a useful notion since ch is clearly additive on short exact sequences and we have the following important result proved in 38 55 Theorem 43 The formal characters of the inequiualent irreducible F5 modules are linearly independent Given 1 i1 in E I de ne its weight wti to be the tuple 39y wk61 where W counts the number of i r 1 n that equal j Thus 39y is an element of the set D of I tuples of non negative integers summing to n Clearly i 1 E I lie in the same Sn orbit under the obvious action by place permutation if and only if wti wt1 hence Ln parametrizes the Sn orbits on I For 39y 6 D and an FSn module M we let M M Z M i 3 13961quot with wti39y Unlike the the subspaces MM are actually FSn submodules of M Indeed as an elementary consequence of Theorem 41 and Lemma 42 we have Lemma 44 The decomposition M 7614 MM is precisely the decom position ofM into blocks as an FSn module We will say that an FSn module M belongs to the block 39y if M 42 Induction and restriction operators Now that we have the notion of formal character we can introduce the i restriction and i induction oper ators e and flu Suppose that 39y 6 Ln Let 39y i 6 D be the tuple 60161 with 67 W for j 7 i and 6 y 1 Similarly assuming this time that y gt 0 let 39y 7i 6 114 be the tuple 60161 with 67 W for j 7 i and 6i jg 4 1 If M is an FSn module belonging to the block 39y 6 Ln de ne eiM resizilM 39y 7 interpreted as 0 in case 39yi 0 4 fM ind1M39y i 5 Extending additively to arbitrary FSn modules M using Lemma 44 and making the obvious de nition on morphisms we obtain exact functors ei FSn mod a FSnA mod and fi FSn mod a FSnH mod The de nition implies Lemma 45 For an FSn module M we have resizilM e EB QM ind Z1M e EB fiM i39eI i39eI Note that eiM can be described alternatively as the generalized eigenspace of an acting on M corresponding to the eigenvalue i This means that the effect of e on characters is easy to describe if chMZaei then cheM Z amwmlwi 6 61quot Ema Let us also mention that there are higher divided power functors 5E7 f5 for each r 2 1 To de ne them start with an FSn module M belonging to the block 39y Let yi7 yii i r times and de ne vii similarly assuming y 2 T View M instead as an FSn x ST module by letting 5 act trivially Embedding Sn x S into Sn in the obvious way we then de ne MM indiztwm m lt7 Extending additively we obtain the functor 127 FSn mod a FSnH mod This exact functor has a two sided adjoint 5E7 FSnH mod a FSn mod It is de ned on a module M belonging to block 39y by eETM MST39y 7 2 7 interpreted as zero if m lt r 8 where MST denotes the space of xed points for the subgroup S lt Sn that permutes 71 1 n r viewed as a module over the subgroup Sn lt Sn that permutes 1 n The following lemma relates the divided power 7 functors 5 and fl to the original functors 5139 12 Lemma 46 For an FSn module M we have ezM g SETMT7 g fiTM7 l39 The functors 5E and 127 have been de ned in an entirely different way by Grojnowski 9 81 which is the key to proving their properties including Lemma 46 43 The af ne KanMoody algebra Let Rn denote the character ring of F5 ie the free Z module spanned by the formal characters of the ir reducible FSn modules In view of Theorem 43 the map ch induces an isomorphism between Rn and the Grothendieck group of the category of all nite dimensional FSn modules Similarly let R denote the Z submodule of Rn spanned by the formal characters of the projective indecomposable FSn modules This time the map ch induces an isomorphism between R and the Grothendieck group of the category of all nite dimensional projec tive FSn modules Let REBR REBRQR 9 n20 n20 The exact functors 5i and fi induce Z linear operators on R Since induction and restriction send projective modules to projective modules Lemma 45 implies that 5139 and 1 do too Hence Pi Q R is invariant under the action of 5139 and flu Extending scalars we get C linear operators e and f on RC C 8 C Z Ff There is also a non degenerate symmetric bilinear form on BC the usual Cartan pairing with respect to which the characters of the projective indecomposables and the irreducibles form a pair of dual bases Theorem 47 The operators ei and E I on RC satisfy the de ning relations of the C hevalley generators of the a ne Kac Moody Lie algebra g of type A2121 resp A00 in case p 0 see 16 Moreover viewing RC as a g module in this way i RC is isomorphic to the basic representation VA0 of g generated by the highest weight vector e0 the character of the irreducible F50 module the decomposition of RC into blocks coincides with its weight space decomposition with respect to the standard C artan subalgebra of g iii the C artan pairing on RC coincides with the Shapovalov form satis fying 607 50 1 iv the lattice 13 C RC is the Z submodule of RC generated by e0 under the action of the operators f5 firr1 E Ir Z 0 the lattice R C RC is the dual lattice to 13 under the Shapovalov form V A lt V This was essentially proved by Lascoux Leclerc Thibon 21 and Ariki 1 for a somewhat different situation and another approach has been given more recently by Grojnowski 9 14210 44 The crystal graph In view of Theorem 47 we can identify BC with the basic representation of the a ine Kac Moody algebra g A2171 Associ ated to this highest weight module Kashiwara has de ned a purely combi natorial object known as a crystal see eg 18 for a survey of this amazing theory We now review the explicit description of this particular crystal due originally to Misra and Miwa 26 This contains all the combinatorial notions we need to complete our exposition of the representation theory Let A A1 2 A2 2 be a partition We identify A with its Young diagram A rs E Zgt0 X Zgt0 s AT Elements r s E Zgt0 x Zgt0 are called nodes We label each node A r s of A with its residue res A E I de ned so that resA E s 7 r mod p see Example 48 below Let i E I be some xed residue A node A E A is called i removable for A if R0 res A i and A 7 A is the diagram of a partition Similarly a node B A is called i addable for A if A0 res B i and A U B is the diagram of a partition Now label all i addable nodes of the diagram A by and all i removable nodes by 7 The i signature of A is the sequence of pluses and minuses obtained by going along the rim of the Young diagram from bottom left 5 to top right and reading off all the signs The reduced i signature of A is obtained from the i signature by successively erasing all neighbouring pairs of the form 7 Example 48 Let p 3 and A 1110995 1 The residues are as follows The 2 addable and 2 removable nodes are as labelled in the diagram Hence the 2 signature of A is 7 7 and the reduced 2 signature is 7 the nodes corresponding to the reduced 2 signature have been circled in the above diagram Note the reduced i signature always looks like a sequence of 7s followed by 7 s Nodes corresponding to a 7 in the reduced i signature are called i normal nodes corresponding to a are called i conormal The leftmost i normal node corresponding to the leftmost 7 in the reduced i signature is called i good and the rightmost i conormal node corresponding to the rightmost in the reduced i signature is called i cogood We recall nally that a partition A is called p regular if it does not have p non zero equal parts It is important to note that if A is p regular and A is an i good node then A 7 A is also p regular Similarly if B is an i cogood node then A U B is p regular By 26 the crystal graph associated to the basic representation VA0 of Q can now be realized as the set of all p regular partitions with a directed edge A L 1 of color i E I if u is obtained from A by adding an i cogood node equivalently A is obtained from u by removing an i good node An example showing part of the crystal graph for p 2 is listed below 45 The modular branching graph Now we explain the relationship between the crystal graph and representation theory The next lemma was rst proved in 20 and in a different way in 11 Q 0 El 1 I V V E 0 1 HE E lt H 1 Lemma 49 Let D be an irreducible FSn module andi E I Then the module eiD resp is either zero or else is a self dual FSnil resp FSn1 module with irreducible socle and head isomorphic to each other Introduce the crystal operators i for an irreducible FSn module D7 let iD socleelD7 fiD soclefiD 10 In View of Lemma 497 iD and fiD are either zero or irreducible Now de ne the modular branching graph the vertices are the isomorphism classes of irreducible FSn modules for all n 2 O7 and there is a directed edge D L E of color i if E E fiD equivalently by F robenius reciprocity7 D E iE The fundamental result is the following Theorem 410 The modular branching graph is uniquely isomorphic as an I colored directed graph to the crystal graph of This theorem was rst stated in this way by Lascoux7 Leclerc and Thi bon 21 they noticed that the combinatorics of Kashiwara s crystal graph as described by Misra and Miwa 26 is exactly the same as the modular branching graph rst determined in 19 A quite different and independent proof of Theorem 410 follows from the more general results of 7 Theorem 410 has some important consequences To start with it implies that the isomorphism classes of irreducible FSn modules are parametrized by the vertices in the crystal graph ie by p regular partitions For a p regular partition A of n we let DA denote the corresponding irreducible FSn module To be quite explicit about this labelling choose a path a A I A in A in the crystal graph from the empty partition to A for 21 in E I Then DA nn ln a 11 where D 3 denotes the irreducible FSo module Note the labelling of the irreducible module DA de ned here is known to agree with the standard labelling of James 13 although James7 construction is quite different Let us state one more result about the structure of the modules eiDA and fiDA see 2 Theorems E E for this and some other more detailed results Theorem 411 Let A be a p Tegular partition of n i Suppose that A is an i Temouable node such that a A 7 A is p Tegular Then eiDA D is the number oft normal nodes to the right ofA counting A itself or 0 ifA is not i normal ii Suppose that B is an i addable node such that 1 A U B is p regular Then fiDA Dquot is the number of i conormal nodes to the left of B counting B itself or 0 if B is not i conormal 46 More on characters Let M be an FSn module De ne maxr 2 0 ezM 344 0 maxr 2 0 M 344 0 12 Note can be computed just from knowledge of the character of M it is the maximal r such that 4 appears with non zero coefficient in ch M Less obViously can also be read off from the character of M By additiVity of fi we may assume that M belongs to the block 39y 6 Ln Then 5130 i 2 YH W41 13 see 9 126 We note the following extremely useful lemma from 11 see also 9 9 Lemma 412 Let D be an irreducible FSn module 6 iDLp LpiD Then egg g gin 105W g 13 The lemma implies that 613 maxr 2 0 313 7r 0 D maxr 2 0 n 7r 0 Thus 64D can also be read off directly from the combinatorics if D DA then SAD is the number of 77s in the reduced i signature of A Similarly LpiltDgt is the number of s in the reduced i signature of A Now we can describe an inductive algorithm to determine the label of an irreducible FSn module D purely from knowledge of its character ch D Pick 8 i E I such that e telD is non zero Let E eESD7 an irreducible ESTHE module with explicitly known character thanks to Lemmas 4127 46 and By induction7 the label of E can be computed purely from knowledge of its character7 say E DA Then7 D ffE D where u is obtained from A by adding the rightmost e of the i conormal nodes We would of course like to be able to reverse this process given a p regular partition A of n we would like to be able to compute the character of the irreducible FSn module DA One can compute a quite effective lower bound for this character inductively using the branching rules of Theorem 411 But only over C is this lower bound always correct indeed ifp 0 then D is equal to the Specht module SA and ch SA Z ei1quotquot iquot 14 i1in summing over all paths 9 I A in the characteristic zero crystal graph aka Young s partition lattice from Q to A Reducing the residues in 14 modulo p in the obvious way gives the formal characters of the Specht module in characteristic p We refer to 32 for a concise self contained approach to the complex representation theory of Sn along the lines described here Now we explain how Lemma 412 can be used to describe some compo sition factors of Specht modulesithis provides new non trivial information on decomposition numbers which is difficult to obtain by other methods The following result follows easily from Lemma 412 Lemma 413 Let M be an FSn module and set 6 If eESM De m gt 0 then ne 7g 0 and M ne m Example 414 Let p 3 By 137 Tables7 the composition factors of the Specht module SGgt4gt2gt1 are Dull7 D9gt47 D9gt227 Dam7 D6gt5gt27 D6gt4gt37 and D6gt4gt2gt17 all appearing with multiplicity 1 As 81S6 4 22 1 by 14 reduced modulo 3 and elS6gt4gt22 56gt4gt2gt17 application of Lemma 413 implies that the following composition factors appear in SGgt4gt22 with multiplicity 1 130242 Dlt9gt4gt1gt Dlt9gt3gt2gt Dlt8gt4gt2gt Dlt62gt2 130142 and D642239 Given 1 i17 7 E I we can gather consecutive equal terms to write it in the form 2jlquot1 m lt15 where jg 7 j51 for all 1 g s lt T For example 2272171 2312 Now7 for an FSn module M7 the tuple 15 is called extremal if 1 e M mg mg 6j5ejs1 for all s r r 7 1 1 lnformally speaking this means that among all the n tuples 1 such that 344 0 we rst choose those with the longest 9 jT string in the end then among these we choose the ones with the longest jPl string preceding the jT string in the end etc By de nition 7 0 if i is extremal for M Example 415 The formal character of the Specht module 55gt2 in char acteristic 3 is 50210201 250120201 250212021 450122021 50212010 250122010 50120210 40120120 The extremal tuples are 0122021 0122010 0120210 and 0120120 Our main result about extremal tuples is Theorem 416 Leti i1 in be an extremal tuple for an irreducible FSn module Dquot Then Dquot 10 121139 and dim DAM m1 quotmel In particular the tuple i is not extremal for any irreducible De gs Dquot Proof We apply induction on r If r 1 then by considering possible n tuples appearing in the Specht module S of which D is a quotient we conclude that n 1 and D D0 So for r 1 the result is obvious Let r gt 1 By de nition of an extremal tuple m 67DA So in View of Lemmas 46 and 412 we have me A7 Mme ej D impej D Moreover n1 jznffl is clearly an extremal tuple for the irreducible module 27 D So the inductive step follows III Corollary 417 IfM is an FSn module andi i1in jin1 H317 is an extremal tuple for M then the multiplicity of DA fin 1 ng as a composition factor of M is dim Mim1l HmTl We note that for any tuple 1 represented in the form 15 and any F5 module M we have that dim is divisible by m1 mrl This follows from the properties of the principal series modules Kato modules for degenerate af ne Hecke algebras see 11 for more details Example 418 In view of Corollary 417 extremal tuple 0122021 in Ex ample 415 yields the composition factor Dlt5gt2l of 5amp2 while the extremal tuple 0120120 yields the composition factor D It turns out that these are exactly the composition factors of 5amp2 see eg 13 Tables For more non trivial examples let us consider a couple of Specht modules for n 11 in characteristic 3 For 6342 Corollary 417 yields composition factors Dlt6gt3gt12 Dlt7gt3gt1 and Dlt8gt2gt1l but misses D01 and for 4332 we get hold of M4322 Dlt5gt3gt2gt1 Dlt8gt2gt1 and Dlt8gt3 but miss 21301 and Dlt5gt4gt12 cf 13 Tables We record here one other useful general fact about formal characters which follows from the Serre relations satis ed by the operators ei 10 Lemma 419 Let M be an FSn module Assume tji1 quot512 E I and i 7 j i Assume that 7 gt 1 Then for any 1 g r g n 7 2 we have dim MWh 7ir7i7j7ir17 397in72 dimMWh7ir7j7i7ir177in72l ii Assume that 7j 1 andp gt 2 Then for any 1 g r g n7 3 we have 2dimMi17397i77i7jvi7ir1739397in73l dimMKilv7irvi7i7jvir177in73l dimMi17 397iT7j7i7i7iT17 7in73l39 iii Assume that 7j 1 andp 2 Then for any 1 g r g n7 4 we have dimMi1iriiijir1in4 3dimMi1i7ijiiir1in4 dimMWl j 1 n4 3dim M2 1 mam awn 47 Blocks Finally we discuss some properties of blocks assuming now that p 7 0 In View of Theorem 47ii the blocks of the F5 for all n are in 171 correspondence with the non zero weight spaces of the basic module VA0 ofg A2121 So let us begin by describing these following 16 ch12 Let P id ZAi 63 Z6 denote the weight lattice associated to g Let oz E I be the simple roots of g de ned from 040 2A0 7 A1 7 A1771 6 04139 7 Ai1 7 Aiil f There is a positive de nite symmetric bilinear form on R Z P with respect to which 040 ozp1A0 and A0 Ap16 form a pair of dual bases Let W denote the Weyl group of g the subgroup of GLGR Z P generated by si 6 I where si is the re ection in the hyperplane orthogonal to 04 Then by 16 1261 the weight spaces of VA0 are the weights wA07d6w lV 16220 For a weight of the form wAO 7 16 we refer to wAO as the corresponding maximal weight and d as the corresponding depth There is a more combinatorial way of thinking of the weights Following 24 11 ex8 and 141 27 to a p regular partition A one associates the corresponding p core A and p weight d A is the partition obtained from A by successively removing as many hooks of length p from the rim of A as possible in such a way that at each step the diagram of a partition remains The number of p hooks removed is the p weight d of A The p cores are in 171 correspondence with the maximal weights ie the weights belonging to the W orbit WAO and the p weight corresponds to the notion of depth 11 introduced in the previous paragraph see 21 53 and 22 2 for the details Now Theorem 47 ii gives yet another proof of the Nakayama conjecture the FSn modules DA and DF belong to the same block if and only if and a have the same p core We will also talk about the p weight of a block B namely the p weight of any such that D belongs to B The Weyl group W acts on the g module RC from 43 the generator si E I of W acting by the familiar formula si exp7ei expfi exp7ei The resulting action preserves the Shapovalov form and leaves the lattices R and Pi invariant Moreover W permutes the weight spaces of BC in the same way as its de ning action on the weight lattice P Since W leaves 6 invariant it follows that the action is transitive on all weight spaces of the same depth So using Theorem 47iii we see Theorem 420 Let B and B be blocks of symmetric groups with the same p weight Then B and B are isometric in the sense that there is an iso morphism between their Grothendieck groups that is an isometry with respect to the C artan form The existence of such isometries was rst noticed by Enguehard lm plicit in Enguehard s paper is the following conjecture made formally by Rickard blocks B and B of symmetric groups with the same p weight should be deriued equiualent This has been proved by Rickard for blocks of p weight 5 Moreover it is now known by work of Marcus 25 and Chuang Kessar 6 that the famous Abelian Defect Group Conjecture of Broue for symmetric groups follows from the Rickard s conjecture above There is one situation that is particularly straightforward when there is actually a Morita equivalence between blocks of the same p weight This is a theorem of Scopes 34 though we are stating the result in a more Lie theoretic way following 22 8 Theorem 421 Let AA 04139 A pa be an ai string of weights of VA0 so A 7 04139 and A r 1Oq are not weights of VA0 Then the functors f5 and 5E de ne mutually inuerse Morita equiualences between the blocks parametrized by A and by A T0412 Proof Since 5E and fl are both left and right adjoint to one another it suffices to check that 5 and f5 induce mutually inverse bijections between the isomorphism classes of irreducible modules belonging to the respective blocks This follows by Lemma 412 El Let us end the discussion with one new result here we can in fact explic itly compute the determinant of the Cartan matrix of a block The details of the proof will appear in Note in view of Theorem 420 the determinant of the Cartan matrix only depends on the p weight of the block Moreover by Theorem 47iii we can work instead in terms of the Shapovalov form 12 on VA0 Using the explicit construction of the latter module over Z given in 7 we show Theorem 422 Let B be a block ofp weight d of F5 Then the determi nant of the Cartan matrix of B is p where N7 r1r2ltp72r1gtltp72r2gt A1T12T2Hi p71 Tl 72 REFERENCES 1 S Ariki On the decomposition numbers of the Hecke algebra of type Cm 171 J Math Kyoto Univ 36 1996 7897808 J Brundan and A Kleshchev Translation functors for general linear and symmetric groups Proc London Math Soc 80 2000 757106 J Brundan and A Kleshchev Projective representations of symmetric groups via Sergeev duality Math Z 239 2002 27768 J Brundan and A Kleshchev Hecke Cli ord superalgebras crystals of type Ag and modular branching rules for Sn Represent Theory 5 2001 3177403 J Brundan and A Kleshchev Cartan determinants and Shapovalov forms to appear in Math Ann 2002 J Chuang and R Kessar Symmetric groups Wreath products Morita equivalences and Brou s abelian defect group conjecture Bull London Math Soc 34 2002 1747184 C De Concini V Kac and D Kazhdan BosonF ermion correspondence over Z in In nitedimensional Lie algebras and groups LuminyMarseille 1988 Adv Ser Math Phys 7 1989 1247137 M Enguehard lsom tries parfaites entre blocs de groupes symetriques Ast risqne 181 182 1990 1577171 1 Grojnowski Af ne 57p controls the modular representation theory of the symmetric group and related Hecke algebras preprint 1999 1 Grojnowski Blocks of the cyclotomic Hecke algebra preprint 1999 1 Grojnowski and M Vazirani Strong multiplicity one theorem for af ne Hecke algebras of type A Transf Groups 6 2001 1437155 J F Humphreys Blocks of projective representations of the symmetric groups J London Math Soc 33 1986 4414152 13 G D James The representation theory of the symmetric groups Lecture Notes in Math vol 682 SpringerVerlag 1978 G James and A Kerber The Representation Theory of the Symmetric Groups AddisonWesley London 1980 A Jucys Symmetric polynomials and the center of the symmetric group ring Report Math Phys 5 1974 1077112 V Kac In nite dimensional Lie algebras Cambridge University Press third edition 995 SJ Kang Crystal bases for quantum af ne algebras and combinatorics of Young Walls preprint Seoul National University 2000 M KashiWara On crystal bases in Representations of groups Ban 1994 CMS Conf Proc 16 1995 1557197 A Kleshchev Branching rules for modular representations of symmetric groups 11 J reine angew Math 459 1995 1637212 A Kleshchev Branching rules for modular representations of symmetric groups 111 Some corollaries and a problem of Mullineux J London Math Soc 54 1996 25738 EEEE E 3 EE HH HO 2 iii g dg U 2 21 13 A Lascoux7 B Leclerc and JY Thibon Hecke algebras at roots of unity and crystal bases of quantum af ne algebras Comm Math Phys 181 1996 2057263 B Leclerc and H Miyachi Some closed formulas for canonical bases of Fock spaces preprint 2001 B Leclerc and JY Thibon q Deformed Fock spaces and modular representations of spin symmetric groups7 J Phys A 30 1997 616376176 1 G Macdonald7 Symmetric functions and Hall polynomials Oxford Mathematical Monographs second edition CUP 1995 A Marcus On equivalences between blocks of group algebras reduction to the simple components J Algebra 184 1996 372396 K Misra and T Miwa Crystal base for the basic representation of Uq 5ri Comm Math Phys 134 1990 79788 AO Morris7 The spin representations of the symmetric group Canad J Math 17 1965 543549 G Murphy A new construction of Young s seminormal representations of the sym metric groups7 J Algebra 69 1981 2877297 G Murphy The idempotents of the symmetric group and Nakayama s conjecture J Algebra 81 1983 2587265 M Nazarov Young s orthogonal form of irreducible projective representations of the symmetric group J London Math Soc 42 1990 4377451 M Nazarov Young s symmetrizers for projective representations of the symmetric group Advances Math 127 1997 1907257 A Okounkov and A Vershik A new approach to representation theory of symmetric groups Selecta Math NS 2 1996 581405 I Schur7 Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen J Reine Angew Math 139 1911 1557 J Scopes7 Cartan matrices and Morita equivalence for blocks of the symmetric groups J Algebra 142 19917 4417455 A N Sergeev7 Tensor algebra of the identity representation as a module over the Lie superalgebras GLrim and Math USSR Sbornik 51 1985 4197427 6 A N Sergeev7 The Howe duality and the projective representations of symmetric groups7 Represent Theory 3 1999 4167434 J Stembridge Shifted tableaux and the projective representations of symmetric groups7 Advances in Math 74 1989 877134 M Vazirani Irreducible modules over the a ine Hecke algebra a strong multiplicity one result PhD thesis7 UC Berkeley 1999

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