The tilting tensor product theorem and decomposition numbers for symmetric groups

We show how the tilting tensor product theorem for algebraic groups implies a reduction formula for decomposition numbers of the symmetric group. We use this to prove generalisations of various theorems of Erdmann and of James and Williams. Let G be a semisimple, connected, simply-connected algebraic group over an algebraically closed field k of characteristic p > 0. The simple modules L(λ) are labelled by dominant weights, and by Steinberg’s tensor product theorem [24] it is sufficient to determine only a finite number of simple modules (those labelled by p-restricted weights). For sufficiently large p these are known: by work of Andersen et al [2] they are given by the Lusztig conjecture [19]. In [6] Donkin has introduced the notion of a tilting module for G (following Ringel [21]). The indecomposable tilting modules T (λ) are also labelled by dominant weights. There is also a tensor product theorem for tilting modules (which will be key in what follows), however it no longer reduces the problem to the study of a finite set of tilting modules. Indeed, the determination of tilting modules is considerably harder than that of simples; there are not even conjectural solutions. The representation theory of the symmetric group Σd over k is also poorly understood. For each partition λ there is a Specht module Sλ, and determining decomposition numbers of these would be sufficient to determine the simples. However little is known except in the case of two part partitions [10], and certain three part partitions [13]. Erdmann has shown [8] that the determination of the decomposition numbers associated to Specht modules indexed by n part partitions is equivalent (by Ringel duality) to determining the good filtration multiplicities of tilting modules for Schur algebras associated to GLn, provided that p > n. The principal aim of this note is to illustrate how tilting module results translate to the symmetric group setting, and in particular to prove a generalisation of results of James and Williams [13] and Erdmann [8, 9] by using the tensor product theorem for tilting modules. ∗ Supported by Nuffield grant scheme NUF-NAL 02. Preliminary work on this paper was undertaken at the Isaac Newton Institute as part of the programme on Symmetric Functions and Macdonald Polynomials c © 2011 Kluwer Academic Publishers. Printed in the Netherlands. kresubmit.tex; 26/09/2011; 18:34; p.1