1 6 M ay 2 00 2 Decomposition of tensor products of modular irreducibles for SL 2

We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group SL2, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed. Introduction We study the structure of L ⊗ L where L,L are simple modules for the algebraic group SL2 = SL2(k) over an algebraically closed field k of positive characteristic p. The solution to this problem is well-known and easily obtained in characteristic zero, but in positive characteristic the problem is significant. Given L ⊗ L, the initial question is to describe its indecomposable direct summands. This is answered in Theorem 2.1. It turns out that each such direct summand is expressible as a twisted tensor product of certain “small” indecomposable tilting modules where the structure of the latter is completely understood (see Lemma 1.1). We note that for p = 2 the module L ⊗ L is always indecomposable, in contrast to what happens for p odd. The indecomposable summands themselves are always contravariantly selfdual, with simple socle (and head), and they occur as subquotients of tilting modules (see Theorem 2.7). On the other hand, each tilting module occurs