Multiplicities of second cell tilting modules

Abstract

We consider three related representation theories: that of a quantum group at a complex root of unity, that of an almost simple algebraic group over an algebraically closed field of prime characteristic and that of the symmetric group. The main results of this paper concern multiplicities in modular tilting modules. We prove a formula, valid for type A n ⩾ 2 , D n , E 6 , E 7 , E 8 and G 2 , giving the multiplicities of indecomposable tilting modules with highest weight in an explicitly described set of alcoves. The proof relies on “quantizations” of the modular tilting modules, and is an application of a recent result by Soergel describing the quantum tilting modules in terms of Hecke algebra combinatorics. In fact the set of alcoves just mentioned corresponds to the second largest Kazhdan–Lusztig cell in the affine Weyl group associated to our root system, giving rise to the phrase “second cell tilting modules.” The Groethendieck group comes with two bases: that of Weyl modules and that of tilting modules. Based on the multiplicity formula we give the coefficients of second cell tilting modules in any Weyl module. Of independent interest is an application of the modular multiplicity formula: we determine the dimensions of a set of simple representations of the symmetric group over a field of characteristic p. The dimension formula covers simple modules parametrized by partitions ( n 1 , … , n n ) with either n 1 − n n − 1 < p − n + 2 or n 2 − n n < p − n + 2 . This generalizes a result of Mathieu [Lett. Math. Phys. 38 (1) (1996) 23–32] as well as a recent result by Jensen and Mathieu [On three lines representations of the symmetric group, in preparation]. Further, it proves in part a conjecture by Mathieu [in: Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama–Kyoto, 1997, Math. Soc. Japan, Tokyo, 2000, pp. 145–212].