Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mccme.ru/tg2007/talks/Suprunenko.pdf Дата изменения: Tue Nov 27 18:12:12 2007 Дата индексирования: Tue Oct 2 12:40:02 2012 Кодировка:

I. D. Suprunenko Institute of Mathematics, National Academy of Sciences of Belarus Minsk, Belarus suprunenko@im.bas-net.by

Lower estimates for the maximal weight multiplicities in irreducible representations of the classical algebraic groups
The goal of this talk is to discuss some results on lower estimates for the maximal weight multiplicities in irreducible representations of the classical algebraic groups in positive characteristic obtained in a course of a joint research with A.A. Baranov and A.A. Osinovskaya. For the groups of types Bn , Cn and Dn such estimates are found under some mild restrictions on the ground field characteristic that do not depend upon n. For the groups of type An at present relevant estimates are obtained only for n large enough with respect to the characteristic. In what follows K is an algebraically closed field of characteristic p > 0; G is a classical algebraic group of rank n over K ; Irr is the set of all rational irreducible representations (or simple modules) of G up to equivalence, Irrp Irr is the subset of p-restricted ones; (M ) is the highest weight of a simple module M , wdeg M is the maximal dimension of a weight subspace in M ; 1 . . . , n are the fundamental weights of G labeled in a standard way. Recall n that a weight i=1 ai i is p-restricted if 0 ai < p for all i. For the classical algebraic groups the simple modules M with wdeg M = 1 were classified in [1, 3]. That result was used in the description of the maximal subgroups of such groups in [1]. Denote by the set of dominant weights of G such that wdeg M = 1 for a simple G-module M with highest weight and by p the subset of all p-restricted weights in . Assume that p = 2 for G = An (K ). Then for n 4 {0; k , 1 k n; (p - 1 - a)k + ak+1 , 0 k n, 0 a p - 1} for G = An (K ), {0, 1 , n } for G = Bn (K ), p = {0, 1 , p-1 n , n-1 + p-3 n } for G = Cn (K ), 2 2 {0, 1 , n-1 , n } for G = Dn (K ), and the weight = j =0 pj j with p-restricted j lies in if and only if all j p . Here for G = An (K ) assume that 0 = n+1 = 0. Theorem 1. Let n = 4k + r 8 with integer k and r and 0 r < 4. Assume that G = Bn (K ), Cn (K ) or Dn (K ), M Irr and (M ) . Suppose that p > 2 for G = Bn (K ) / or Dn (K ) and p > 7 for G = Cn (K ). Then wdeg M n - 4 - r n - 7. One easily concludes that wdeg M = n if p > 2, n > 2 for G = Bn (K ) and > 3 for G = Dn (K ), M Irr and (M ) = 2 for G = Bn (K ) or Dn (K ) and 21 for G = Cn (K ). Hence the estimates in Theorem 1 are asymptotically exact. For groups of type An the situation is substantially more difficult. For each positive integer a and n large enough with respect to a there exist p-restricted An (K )-modules M d with wdeg M > a, but small enough with respect to n. Indeed, if (M ) = i=1 ai i or d i=1 ai n-i+1 , then for large n the parameter wdeg M is b ounded by some constant that depends on a1 , . . ., ad only and does not depend on n. For this type one has to take into account the polynomial degree of an irreducible module M when trying to estimate wdeg M . n n If (M ) = i=1 ai i , set pdeg M = i=1 iai . Below M is the module dual to M . Irr / Theorem 2. Let G = An (K ), M p , M p , and n > 4p2 . Assume that pdeg M > n and pdeg M > n. Then wdeg M > n/p - 1.
s

1


For G = An (K ) we also suppose that a stronger (may be, linear) estimate holds for wdeg M if both pdeg M and pdeg M > n. Some results on wdeg M for tensor decomposable irreducible modules M will be discussed as well. Estimates of weight multiplicities indicated above and expected results in this direction for groups of type An can be used for recognizing linear groups containing matrices with small eigenvalue multiplicities. Indeed, for groups of types Bn , Cn , and Dn it occurs (may be, under some restrictions on the characteristic) that the images of almost all their representations do not contain matrices all whose eigenvalue multiplicities are small enough with respect to the group rank, only the representations lying in certain well-defined classes yield exceptions. These estimates enable us to classify inductive systems of representations with totally bounded weight multiplicities for natural embeddings of classical groups. Inductive systems of representations have been introduced by A.E. Zalesskii in [2]. They yield an asymptotic version of the branching rules for relevant embeddings and can be applied to the study of ideals in group algebras of locally finite groups. This research was done in the framework of the pro ject "Asymptotic problems in representation theory" supported by the Royal Society for Excellence in Science. A.A. Osinovskaya and I.D. Suprunenko were supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the State Basic Research Programmes "Mathematical Structures" (2001­2005) and "Mathematical models" (2006­2010). References [1] G.M. Seitz, The maximal subgroups of classical algebraic groups, Memoirs of the AMS, 365 (1987), 1­286. [2] A.E. Zalesskii, Group rings of locally finite groups and representation theory, In Proceedings of the International Conference on Algebra, Novosibirsk, 1989; Contemporary Math. 131 (1992), part 1, 453­472. [3] A.E. Zalesskii and I.D. Suprunenko, Representations of dimensions (pn ± 1) of a symplectic group of degree 2n over a finite field (in Russian), Vestsi AN BSSR, Ser. Fiz.-Mat. Navuk, no. 6 (1987), 9­15.

2