Decomposition Matrix of GL(n,q) and the Heisenberg algebra

Transcription

1 Decomposition Matrix of GL(n,q) and the Heisenberg algebra Bhama Srinivasan University of Illinois at Chicago mca-13, August 2013 Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

2 G is a nite group. Ordinary representation of G : Representation over a eld of characteristic 0 Modular representation of G : Representation over a eld of characteristic p, p divides jg j. The character of a representation of G over an algebraically closed eld of characteristic 0 (e.g. C) is an "ordinary" character. Brauer character of a modular representation: a complex-valued function on the p-regular elements of G. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

3 p a prime integer K a suciently large eld of characteristic 0 O a complete discrete valuation ring with quotient eld K k residue eld of O, char k=p hama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

4 A representation of G over K is equivalent to a representation over O, and can then be reduced mod p to get a modular representation of G over k. Thus: Compare ordinary and p-modular (Brauer) characters. The decomposition matrix D (over Z) is the transition matrix between ordinary and Brauer characters. Entries of D are decomposition numbers. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

5 In KG -module M, nd lattice, O-module. reduce mod p to get kg -module M D computes composition factor multiplicities of simple kg -modules, in reduction of simple KG or O-module reduced mod p. The matrix D is our main object of study. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

6 G n = GL(n; q), ` a prime not dividing q, e the order of q mod `. Unipotent representations of G n Borel) and are indexed by partitions of n. Example: Steinberg representation are constituents of Ind Gn B (1) (B a End(Ind Gn B (1)) is isomorphic to the Hecke algebra H n of type A. Problem: Unipotent part of Decomposition Matrix of G n. Surprise: The matrix is square. (Fong-BS) Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

7 Dene S n, the q-schur algebra, endomorphism algebra of a sum of permutation representations of the Hecke algebra H n of type A. S n dened over eld of characteristic 0 or ` Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

8 S n G n has Weyl modules, irreducible modules has Specht modules (for unipotent representations), irreducible modules Entries of D are composition multiplicities in either case. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

9 S n over k of characteristic 0, q 2 k, an e-th root of unity. The decomposition matrix is square, has entries the multiplicities of irreducibles in Weyl modules. Known by Varagnolo-Vasserot: transition matrix between a Leclerc-Thibon canonical basis, and a standard basis of a Fock space. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

10 S n over k of characteristic `, q 2 k, an e-th root of unity. Again, the decomposition matrix is square, has entries the multiplicities of irreducibles in Weyl modules. There is the square part of the decomposition matrix of G n, rows indexed by unipotent characters, columns by Brauer characters. (Dipper-James) These two matrices are the same! Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

11 S n over k of characteristic `, q 2 k, an e-th root of unity. Again, the decomposition matrix is square, has entries the multiplicities of irreducibles in Weyl modules. There is the square part of the decomposition matrix of G n, rows indexed by unipotent characters, columns by Brauer characters. (Dipper-James) These two matrices are the same! Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

12 Unipotent characters of G n and Weyl modules of S n are both indexed by partitions of n. Leads to Fock space. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

13 Fock space(level 1): F v = Q(v)u, a vector space over Q(v) with basis u indexed by all partitions. Fix a positive integer e. U v ( sl c e ) acts on this space! Generators e i ; f i are functors on the Fock space: i-induction, i-restriction. Not an irreducible representation, thus we need U v ( gl c e ), so the Heisenberg algebra comes in. Lie algebra c gl e = c sl e + h e. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

14 Fock space(level 1): F v = Q(v)u, a vector space over Q(v) with basis u indexed by all partitions. Fix a positive integer e. U v ( sl c e ) acts on this space! Generators e i ; f i are functors on the Fock space: i-induction, i-restriction. Not an irreducible representation, thus we need U v ( gl c e ), so the Heisenberg algebra comes in. Lie algebra c gl e = c sl e + h e. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

15 The algebra H e has generators hb k jk 2 Z f0gi with relations [B k ; B`] = k 1 v 2nk 1 v 2k k; ` Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

16 (Leclerc-Thibon) Commuting operators V k (k > 1) in H e acting on F v, used to nd new basis: V k (u ) = X ( q) s(=) u where the sum is over all such that is obtained from by adding k e-ribbons, such that the tail of each ribbon is not upon another ribbon. (ribbon= skew-hook, s is the spin.) More generally: V where is a composition: If = f 1 ; 2 ; : : :g then V = V 1 :V 2 : : :. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

17 (Leclerc-Thibon) Commuting operators V k (k > 1) in H e acting on F v, used to nd new basis: V k (u ) = X ( q) s(=) u where the sum is over all such that is obtained from by adding k e-ribbons, such that the tail of each ribbon is not upon another ribbon. (ribbon= skew-hook, s is the spin.) More generally: V where is a composition: If = f 1 ; 2 ; : : :g then V = V 1 :V 2 : : :. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

18 Now regard F v as a vector space over Q l (v) with basis u indexed by all partitions. A n = category of unipotent representations of G n. A = ( n>0k 0 (A n )) Z Q l (v), isomorphic to F v as a Q l (v)-vector space, since A also has a basis indexed by partitions. H e acts on A by the operators V. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

19 Regard A as having a basis indexed by unipotent characters f g where runs through all partitions. Theorem G n = GL(n; q), L = G n GL(k ; q e ). Dene Lusztig maps L k : K 0 (A n )! K 0 (A n+ke ) by : [ ]! [R G n+ke L ( (k) )]. Then L k coincides with the operator V k Deligne-Lusztig induction. specialized at v = 1. Here R G n+ke L More generally, dene L, a partition of k. This coincides with V. is Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

20 fg n = GL(n; F q ), P = f L n U f G n a parabolic subgroup, L n G n the group of F q -rational points of L f n. X L = Deligne-Lusztig G n -variety-l n, dened as X L = fg 2 G f n jg 1 F (g) 2 Ug. We have a complex R c (X L ), gives rise to a functor D b (Q l L n mod)! D b (Q l G n mod) (Bonnafe-Rouquier, [Publ. Math. IHES 97 (2003), 1-59]) Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

21 Set v = 1. Categorify V k by a Deligne-Lusztig functor. Category B = n>0d b (Q l G n mod) (unipotent representations). Functor S k : D b (Q l G n mod)! D b (Q l G n+ke mod) is dened by: C! R c (X n ) L (C K Ln (k)). K (k) is a complex in D b (Q l (GL(k ; q e ) mod) with one term the trivial representation of GL(k ; q e ), parametrized by the partition (k). Note L n = G n GL(k ; q e ) G n+ke. So S k maps B to B. The functor S k categories V k. Similarly have S. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

22 Remark. Licata and Savage study actions of the Heisenberg algebra on the category we have denoted by A. Their operators are ordinary induction and restriction. Remark. Shan and Vasserot have dened actions of H e on Fock space of higher level by operators analogous to V, L. Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

23 (Geck) Decomposition numbers for GL n (q), ` large known via the q-schur algebra S n Not known: Decomposition numbers for GL n (q), all ` Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21

24 References S.Ariki, Graded q-schur algebras, Preprint, arxiv: R.Dipper, G.James, Proc. London Math. Soc. 59 (1989), B.Leclerc, J-Y. Thibon, Canonical bases of q-deformed Fock spaces, Int. math. Res. Notices 9 (1996), A.Licata, A.Savage, Hecke algebras, nite general linear groups, Heisenberg categorication, arxiv P.Shan, E.Vasserot, Heisenberg algebras and rational DAHA, JAMS 25 (2012), Bhama Srinivasan (University of Illinois at Chicago) Heisenberg algebra mca-13, August / 21