4.1. Formal characters. For k = 1,..., n, we define the Jucys-Murphy element. k 1. i=1

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1 4. The representaton theory of the symmetrc group In ths chapter we gve an overvew of the p-modular representaton theory of the symmetrc group S n and ts connecton to the affne Kac-Moody algebra of type A () p. We work over a feld F of arbtrary characterstc p. 4.. Formal characters. For k =,..., n, we defne the Jucys-Murphy element k x k := ( k) F S n, () = see [5, 28]. It s straghtforward to show that the elements x, x 2,..., x n commute wth one another. Moreover, we have by [5] or [29,.9]: Theorem 4.. The center of the group algebra F S n s precsely the set of all symmetrc polynomals n the elements x, x 2,..., x n. Now let M be an F S n -module. Let I = Z/pZ dentfed wth the prme subfeld of F. For = (,..., n ) I n, defne M[] := {v M (x r r ) N v = for N and each r =,..., n}. Thus, M[] s the smultaneous generalzed egenspace for the commutng operators x,..., x n correspondng to the egenvalues,..., n respectvely. Lemma 4.2. Any F S n -module M decomposes as M = I n M[]. Proof. It suffces to show that all egenvalues of x r on M le n I, for each r =,..., n. Ths s obvous f r = (as x = ). Now assume that all egenvalues of x r on M le n I, and consder x r+. Let v M be a smultaneous egenvector for the commutng operators x r and x r+. Consder the subspace N spanned by v and s r v. Suppose that N s two dmensonal. Then the ( matrx ) for the acton of c x r on N wth respect to the bass {v, s r v} s for some, j I j and c F (by assumpton on the egenvalues ( of x r ). Hence, ) the matrx for j the acton of x r+ = s r x r s r + s r on N s. Snce v was an c + egenvector for x r+, we see that c =, hence v has egenvalue j for x r+ as requred. Fnally suppose that N s one dmensonal. Then, s r v = ±v. Hence, f x r v = v for I, then x r+ v = (s r x r s r + s r )v = ( ± )v. Snce ± I, we are done. We defne the formal character ch M of a fnte dmensonal F S n -module M to be ch M := I n dm(m[])e, (2)

2 2 an element of the free Z-module on bass {e I n }. Ths s a useful noton, snce ch s clearly addtve on short exact sequences and we have the followng mportant result proved n [38, 5.5]: Theorem 4.3. The formal characters of the nequvalent rreducble F S n - modules are lnearly ndependent. Gven = (,..., n ) I n, defne ts weght wt() to be the tuple γ = (γ ) I where γ j counts the number of r (r =,..., n) that equal j. Thus, γ s an element of the set Γ n of I-tuples of non-negatve ntegers summng to n. Clearly, j I n le n the same S n -orbt (under the obvous acton by place permutaton) f and only f wt() = wt(j), hence Γ n parametrzes the S n -orbts on I n. For γ Γ n and an F S n -module M, we let M[γ] := M[]. (3) I n wth wt()=γ Unlke the M[], the subspaces M[γ] are actually F S n -submodules of M. Indeed, as an elementary consequence of Theorem 4. and Lemma 4.2, we have: Lemma 4.4. The decomposton M = γ Γ n M[γ] s precsely the decomposton of M nto blocks as an F S n -module. We wll say that an F S n -module M belongs to the block γ f M = M[γ] Inducton and restrcton operators. Now that we have the noton of formal character, we can ntroduce the -restrcton and -nducton operators e and f. Suppose that γ Γ n. Let γ + Γ n+ be the tuple (δ ) I wth δ j = γ j for j and δ = γ +. Smlarly, assumng ths tme that γ >, let γ Γ n be the tuple (δ ) I wth δ j = γ j for j and δ = γ. If M s an F S n -module belongng to the block γ Γ n, defne e M := (res Sn S n M)[γ ] (nterpreted as n case γ = ), (4) f M := (nd S n+ S n M)[γ + ]. (5) Extendng addtvely to arbtrary F S n -modules M usng Lemma 4.4 and makng the obvous defnton on morphsms, we obtan exact functors e : F S n -mod F S n -mod and f : F S n -mod F S n+ -mod. The defnton mples: Lemma 4.5. For an F S n -module M we have res Sn S n M = I e M, nd S n+ S n M = f M. I Note that e M can be descrbed alternatvely as the generalzed egenspace of x n actng on M correspondng to the egenvalue. Ths means that the

3 effect of e on characters s easy to descrbe: f ch M = I n a e then ch (e M) = I n a (,..., n,)e. (6) Let us also menton that there are hgher dvded power functors e (r), f (r) for each r. To defne them, start wth an F S n -module M belongng to the block γ. Let γ + r = γ (r tmes), and defne γ r smlarly (assumng γ r). Vew M nstead as an F (S n S r )-module by lettng S r act trvally. Embeddng S n S r nto S n+r n the obvous way, we then defne f (r) M := (nd S n+r S n S r M)[γ + r ]. (7) Extendng addtvely, we obtan the functor f (r) : F S n -mod F S n+r -mod. Ths exact functor has a two-sded adjont e (r) : F S n+r -mod F S n -mod. It s defned on a module M belongng to block γ by e (r) M := (M Sr )[γ r ] (nterpreted as zero f γ < r), (8) where M Sr denotes the space of fxed ponts for the subgroup S r < S n+r that permutes n +,..., n + r, vewed as a module over the subgroup S n < S n+r that permutes,..., n. The followng lemma relates the dvded power functors e (r) and f (r) to the orgnal functors e, f : Lemma 4.6. For an F S n -module M we have e r M = (e (r) M) r!, f r M = (f (r) M) r!. The functors e (r) and f (r) have been defned n an entrely dfferent way by Grojnowsk [9, 8.], whch s the key to provng ther propertes ncludng Lemma The affne Kac-Moody algebra. Let R n denote the character rng of F S n,.e. the free Z-module spanned by the formal characters of the rreducble F S n -modules. In vew of Theorem 4.3, the map ch nduces an somorphsm between R n and the Grothendeck group of the category of all fnte dmensonal F S n -modules. Smlarly, let Rn denote the Z-submodule of R n spanned by the formal characters of the projectve ndecomposable F S n -modules. Ths tme, the map ch nduces an somorphsm between Rn and the Grothendeck group of the category of all fnte dmensonal projectve F S n -modules. Let R = n R n, R = n R n R. (9) The exact functors e and f nduce Z-lnear operators on R. Snce nducton and restrcton send projectve modules to projectve modules, Lemma 4.5 mples that e and f do too. Hence, R R s nvarant under the acton of e and f. 3

4 4 Extendng scalars we get C-lnear operators e and f on R C := C Z R = C Z R. There s also a non-degenerate symmetrc blnear form on R C, the usual Cartan parng, wth respect to whch the characters of the projectve ndecomposables and the rreducbles form a par of dual bases. Theorem 4.7. The operators e and f ( I) on R C satsfy the defnng relatons of the Chevalley generators of the affne Kac-Moody Le algebra g of type A () p (resp. A n case p = ), see [6]. Moreover, vewng R C as a g-module n ths way, () R C s somorphc to the basc representaton V (Λ ) of g, generated by the hghest weght vector e (the character of the rreducble F S - module); () the decomposton of R C nto blocks concdes wth ts weght space decomposton wth respect to the standard Cartan subalgebra of g; () the Cartan parng on R C concdes wth the Shapovalov form satsfyng (e, e ) = ; (v) the lattce R R C s the Z-submodule of R C generated by e under the acton of the operators f (r) = f r /r! ( I, r ); (v) the lattce R R C s the dual lattce to R under the Shapovalov form. Ths was essentally proved by Lascoux-Leclerc-Thbon [2] and Ark [] (for a somewhat dfferent stuaton), and another approach has been gven more recently by Grojnowsk [9, 4.2],[] The crystal graph. In vew of Theorem 4.7, we can dentfy R C wth the basc representaton of the affne Kac-Moody algebra g = A () p. Assocated to ths hghest weght module, Kashwara has defned a purely combnatoral object known as a crystal, see e.g. [8] for a survey of ths amazng theory. We now revew the explct descrpton of ths partcular crystal, due orgnally to Msra and Mwa [26]. Ths contans all the combnatoral notons we need to complete our exposton of the representaton theory. Let λ = (λ λ 2 ) be a partton. We dentfy λ wth ts Young dagram λ = {(r, s) Z > Z > s λ r }. Elements (r, s) Z > Z > are called nodes. We label each node A = (r, s) of λ wth ts resdue res A I defned so that res A (s r) (mod p), see Example 4.8 below. Let I be some fxed resdue. A node A λ s called -removable (for λ) f (R) res A = and λ {A} s the dagram of a partton. Smlarly, a node B / λ s called -addable (for λ) f (A) res B = and λ {B} s the dagram of a partton. Now label all -addable nodes of the dagram λ by + and all -removable nodes by. The -sgnature of λ s the sequence of pluses and mnuses obtaned by gong along the rm of the Young dagram from bottom left

5 to top rght and readng off all the sgns. The reduced -sgnature of λ s obtaned from the -sgnature by successvely erasng all neghbourng pars of the form +. Example 4.8. Let p = 3 and λ = (,, 9, 9, 5, ). The resdues are as follows: The 2-addable and 2-removable nodes are as labelled n the dagram: Hence, the 2-sgnature of λ s +,,, + and the reduced 2-sgnature s +, (the nodes correspondng to the reduced 2-sgnature have been crcled n the above dagram). Note the reduced -sgnature always looks lke a sequence of + s followed by s. Nodes correspondng to a n the reduced -sgnature are called -normal, nodes correspondng to a + are called -conormal. The leftmost -normal node (correspondng to the leftmost n the reduced -sgnature) s called -good, and the rghtmost -conormal node (correspondng to the rghtmost + n the reduced -sgnature) s called -cogood. We recall fnally that a partton λ s called p-regular f t does not have p non-zero equal parts. It s mportant to note that f λ s p-regular and A s an -good node, then λ {A} s also p-regular. Smlarly f B s an -cogood node, then λ {B} s p-regular. By [26], the crystal graph assocated to the basc representaton V (Λ ) of g can now be realzed as the set of all p-regular parttons, wth a drected edge λ µ of color I f µ s obtaned from λ by addng an -cogood node (equvalently, λ s obtaned from µ by removng an -good node). An example showng part of the crystal graph for p = 2 s lsted below The modular branchng graph. Now we explan the relatonshp between the crystal graph and representaton theory. The next lemma was frst proved n [2], and n a dfferent way n [].

6 6. Lemma 4.9. Let D be an rreducble F S n -module and I. Then, the module e D (resp. f D) s ether zero, or else s a self-dual F S n - (resp. F S n+ -) module wth rreducble socle and head somorphc to each other. Introduce the crystal operators ẽ, f : for an rreducble F S n -module D, let ẽ D := socle(e D), f D := socle(f D). () In vew of Lemma 4.9, ẽ D and f D are ether zero or rreducble. Now defne the modular branchng graph: the vertces are the somorphsm classes of rreducble F S n -modules for all n, and there s a drected edge [D] [E] of color f E = f D (equvalently by Frobenus recprocty, D = ẽ E). The fundamental result s the followng: Theorem 4.. The modular branchng graph s unquely somorphc (as an I-colored, drected graph) to the crystal graph of 4.4. Ths theorem was frst stated n ths way by Lascoux, Leclerc and Thbon [2]: they notced that the combnatorcs of Kashwara s crystal graph as descrbed by Msra and Mwa [26] s exactly the same as the modular branchng graph frst determned n [9]. A qute dfferent and ndependent proof of Theorem 4. follows from the more general results of [9].

7 Theorem 4. has some mportant consequences. To start wth, t mples that the somorphsm classes of rreducble F S n -modules are parametrzed by the vertces n the crystal graph,.e. by p-regular parttons. For a p-regular partton λ of n, we let D λ denote the correspondng rreducble F S n -module. To be qute explct about ths labellng, choose a path 2 n λ n the crystal graph from the empty partton to λ, for,..., n I. Then, D λ := f n... f D, () where D denotes the rreducble F S -module. Note the labellng of the rreducble module D λ defned here s known to agree wth the standard labellng of James [3], although James constructon s qute dfferent. Let us state one more result about the structure of the modules e D λ and f D λ, see [2, Theorems E, E ] for ths and some other more detaled results. Theorem 4.. Let λ be a p-regular partton of n. () Suppose that A s an -removable node such that µ := λ {A} s p-regular. Then, [e D λ : D µ ] s the number of -normal nodes to the rght of A (countng A tself), or f A s not -normal. () Suppose that B s an -addable node such that ν := λ {B} s p- regular. Then, [f D λ : D ν ] s the number of -conormal nodes to the left of B (countng B tself), or f B s not -conormal More on characters. Let M be an F S n -module. Defne ε (M) = max{r e r M } ϕ (M) = max{r f r M }. (2) Note ε (M) can be computed just from knowledge of the character of M: t s the maxmal r such that e (...,r) appears wth non-zero coeffcent n ch M. Less obvously, ϕ (M) can also be read off from the character of M. By addtvty of f, we may assume that M belongs to the block γ Γ n. Then ϕ (M) = ε (M) + δ, 2γ + γ + γ +, (3) see [9, 2.6]. We note the followng extremely useful lemma from [], see also [9, 9]: Lemma 4.2. Let D be an rreducble F S n -module, ε = ε (D), ϕ = ϕ (D). Then, e (ε) D = ẽ ε (ϕ) D, f D = f ϕ D. The lemma mples that ε (D) = max{r ẽ r D }, ϕ (D) = max{r f r D }. Thus, ε (D) can also be read off drectly from the combnatorcs: f D = D λ, then ε (D) s the number of s n the reduced -sgnature of λ. Smlarly, ϕ (D) s the number of + s n the reduced -sgnature of λ. Now we can descrbe an nductve algorthm to determne the label of an rreducble F S n -module D purely from knowledge of ts character ch D. Pck 7

8 8 I such that ε := ε (D) s non-zero. Let E = e (ε) D, an rreducble F S n ε - module wth explctly known character thanks to Lemmas 4.2, 4.6 and (6). By nducton, the label of E can be computed purely from knowledge of ts character, say E = D λ. Then, D = f εe = D µ where µ s obtaned from λ by addng the rghtmost ε of the -conormal nodes. We would of course lke to be able to reverse ths process: gven a p-regular partton λ of n, we would lke to be able to compute the character of the rreducble F S n -module D λ. One can compute a qute effectve lower bound for ths character nductvely usng the branchng rules of Theorem 4.. But only over C s ths lower bound always correct: ndeed f p = then D λ s equal to the Specht module S λ and ch S λ = e (,..., n) (4) (,..., n) summng over all paths 2 n λ n the characterstc zero crystal graph (a.k.a. Young s partton lattce) from to λ. (Reducng the resdues n (4) modulo p n the obvous way gves the formal characters of the Specht module n characterstc p.) We refer to [32] for a concse self-contaned approach to the complex representaton theory of S n along the lnes descrbed here. Now we explan how Lemma 4.2 can be used to descrbe some composton factors of Specht modules ths provdes new non-trval nformaton on decomposton numbers whch s dffcult to obtan by other methods. The followng result follows easly from Lemma 4.2. Lemma 4.3. Let M be an F S n -module and set ε = ε (M). If [e (ε) M : D µ ] = m > then f εdµ and [M : f εdµ ] = m. Example 4.4. Let p = 3. By [3, Tables], the composton factors of the Specht module S (6,4,2,) are D (2,), D (9,4), D (9,22), D (7,4,2), D (6,5,2), D (6,4,3), and D (6,4,2,), all appearng wth multplcty. As ε (S (6,4,22) ) = (by (4) reduced modulo 3) and e S (6,4,22) = S (6,4,2,), applcaton of Lemma 4.3 mples that the followng composton factors appear n S (6,4,22 ) wth multplcty : D (2,2), D (9,4,), D (9,3,2), D (8,4,2), D (62,2), D (6,42), and D (6,4,22). Gven = (,..., n ) I n we can gather consecutve equal terms to wrte t n the form = (j m... j mr r ) (5) where j s j s+ for all s < r. For example (2, 2, 2,, ) = (2 3 2 ). Now, for an F S n -module M, the tuple (5) s called extremal f m s = ε js (e m s+ j s+... e mr j r M) for all s = r, r,...,. Informally speakng ths means that among all the n-tuples such that M[] we frst choose those wth the longest

9 j r -strng n the end, then among these we choose the ones wth the longest j r -strng precedng the j r -strng n the end, etc. By defnton M[] f s extremal for M. Example 4.5. The formal character of the Specht module S (5,2) n characterstc 3 s e (22) + 2e (22) + 2e (222 ) + 4e (22 2 ) +e (22) + 2e (22 ) + e (22) + e (22). The extremal tuples are (2 2 2 ), (2 2 ), (22), and (22). Our man result about extremal tuples s Theorem 4.6. Let = (,..., n ) = (j m... jr mr ) be an extremal tuple for an rreducble F S n -module D λ. Then D λ = f n... f D, and dm D λ [] = m!... m r!. In partcular, the tuple s not extremal for any rreducble D µ = D λ. Proof. We apply nducton on r. If r =, then by consderng possble n-tuples appearng n the Specht module S λ, of whch D λ s a quotent, we conclude that n = and D = D (). So for r = the result s obvous. Let r >. By defnton of an extremal tuple, m r = ε jr (D λ ). So, n vew of Lemmas 4.6 and 4.2, we have e mr j r D λ = m r!ẽ mr j r D λ. Moreover, (j m... j m r r ) s clearly an extremal tuple for the rreducble module ẽ mr j r D λ. So the nductve step follows. Corollary 4.7. If M s an F S n -module and = (,..., n ) = (j m... j mr r ) s an extremal tuple for M then the multplcty of D λ := f n... f D as a composton factor of M s dm M[]/(m!... m r!). We note that for any tuple represented n the form (5) and any F S n - module M we have that dm M[] s dvsble by m!... m r!. Ths follows from the propertes of the prncpal seres modules ( Kato modules ) for degenerate affne Hecke algebras, see [] for more detals. Example 4.8. In vew of Corollary 4.7 extremal tuple (2 2 2 ) n Example 4.5 yelds the composton factor D (5,2) of S (5,2), whle the extremal tuple (22) yelds the composton factor D (7). It turns out that these are exactly the composton factors of S (5,2), see e.g. [3, Tables]. For more non-trval examples let us consder a couple of Specht modules for n = n characterstc 3. For S (6,3,2), Corollary 4.7 yelds composton factors D (6,3,2), D (7,3,), and D (8,2,) but msses D (), and for S (4,3,22) we get hold of D (4,3,22), D (5,3,2,), D (8,2,), and D (8,3), but mss 2D () and D (5,4,2), cf. [3, Tables]. We record here one other useful general fact about formal characters whch follows from the Serre relatons satsfed by the operators e : 9

10 Lemma 4.9. Let M be an F S n -module. Assume, j,,..., n 2 I and j. () Assume that j >. Then for any r n 2 we have dm M[(,..., r,, j, r+,..., n 2 )] = dm M[(,..., r, j,, r+,..., n 2 )]. () Assume that j = and p > 2. Then for any r n 3 we have 2 dm M[(,..., r,, j,, r+,..., n 3 )] = dm M[(,..., r,,, j, r+,..., n 3 )] + dm M[(,..., r, j,,, r+,..., n 3 )]. () Assume that j = and p = 2. Then for any r n 4 we have dm M[(,..., r,,,, j, r+,..., n 4 )] +3 dm M[(,..., r,, j,,, r+,..., n 4 )] = dm M[(,..., r, j,,,, r+,..., n 4 )] +3 dm M[(,..., r,,, j,, r+,..., n 4 )] Blocks. Fnally we dscuss some propertes of blocks, assumng now that p. In vew of Theorem 4.7(), the blocks of the F S n for all n are n correspondence wth the non-zero weght spaces of the basc module V (Λ ) of g = A () p. So let us begn by descrbng these followng [6, ch.2]. Let P = I ZΛ Zδ denote the weght lattce assocated to g. Let α ( I) be the smple roots of g, defned from α = 2Λ Λ Λ p + δ, α = 2Λ Λ + Λ ( ). (6) There s a postve defnte symmetrc blnear form (..) on R Z P wth respect to whch α,..., α p, Λ and Λ,..., Λ p, δ form a par of dual bases. Let W denote the Weyl group of g, the subgroup of GL(R Z P ) generated by s ( I), where s s the reflecton n the hyperplane orthogonal to α. Then, by [6, (2.6.)], the weght spaces of V (Λ ) are the weghts {wλ dδ w W, d Z }. For a weght of the form wλ dδ, we refer to wλ as the correspondng maxmal weght, and d as the correspondng depth. There s a more combnatoral way of thnkng of the weghts. Followng [24, I., ex.8] and [4, 2.7], to a p-regular partton λ one assocates the correspondng p-core λ and p-weght d: λ s the partton obtaned from λ by successvely removng as many hooks of length p from the rm of λ as possble, n such a way that at each step the dagram of a partton remans. The number of p-hooks removed s the p-weght d of λ. The p-cores are n correspondence wth the maxmal weghts,.e. the weghts belongng to the W -orbt W Λ, and the p-weght corresponds to the noton of depth

11 ntroduced n the prevous paragraph, see [2, 5.3] and [22, 2] for the detals. Now Theorem 4.7() gves yet another proof of the Nakayama conjecture: the F S n -modules D λ and D µ belong to the same block f and only f λ and µ have the same p-core. We wll also talk about the p-weght of a block B, namely, the p-weght of any λ such that D λ belongs to B. The Weyl group W acts on the g-module R C from 4.3, the generator s ( I) of W actng by the famlar formula s = exp( e ) exp(f ) exp( e ). The resultng acton preserves the Shapovalov form, and leaves the lattces R and R nvarant. Moreover, W permutes the weght spaces of R C n the same way as ts defnng acton on the weght lattce P. Snce W leaves δ nvarant, t follows that the acton s transtve on all weght spaces of the same depth. So usng Theorem 4.7() we see: Theorem 4.2. Let B and B be blocks of symmetrc groups wth the same p-weght. Then, B and B are sometrc, n the sense that there s an somorphsm between ther Grothendeck groups that s an sometry wth respect to the Cartan form. The exstence of such sometres was frst notced by Enguehard [8]. Implct n Enguehard s paper s the followng conjecture, made formally by Rckard: blocks B and B of symmetrc groups wth the same p-weght should be derved equvalent. Ths has been proved by Rckard for blocks of p-weght 5. Moreover, t s now known by work of Marcus [25] and Chuang-Kessar [6] that the famous Abelan Defect Group Conjecture of Broué for symmetrc groups follows from the Rckard s conjecture above. There s one stuaton that s partcularly straghtforward, when there s actually a Morta equvalence between blocks of the same p-weght. Ths s a theorem of Scopes [34], though we are statng the result n a more Le theoretc way followng [22, 8]: Theorem 4.2. Let Λ, Λ + α,..., Λ + rα be an α -strng of weghts of V (Λ ) (so Λ α and Λ + (r + )α are not weghts of V (Λ )). Then the functors f (r) and e (r) defne mutually nverse Morta equvalences between the blocks parametrzed by Λ and by Λ + rα. Proof. Snce e (r) and f (r) are both left and rght adjont to one another, t suffces to check that e (r) and f (r) nduce mutually nverse bjectons between the somorphsm classes of rreducble modules belongng to the respectve blocks. Ths follows by Lemma 4.2. Let us end the dscusson wth one new result here: we can n fact explctly compute the determnant of the Cartan matrx of a block. The detals of the proof wll appear n [5]. Note n vew of Theorem 4.2, the determnant of the Cartan matrx only depends on the p-weght of the block. Moreover, by Theorem 4.7(), we can work nstead n terms of the Shapovalov form

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