Primitive central idempotents of finite group rings of symmetric groups

Abstract

Let p be a prime. We denote by S n the symmetric group of degree n, by An the alternating group of degree n and by F p the field with p elements. An important concept of modular representation theory of a finite group G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring F q G, where q is a prime power. Here, we describe a new method to compute the primitive central idempotents of F q G for arbitrary prime powers q and arbitrary finite groups G. For the group rings F p S n of the symmetric group, we show how to derive the primitive central idempotents of F P S n-P from the idempotents of F p S n . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of F p S n which contains the primitive central idempotents. The described results are most efficient for p = 2. In an appendix we display all primitive central idempotents of F 2 S n and F 4 A n for n ≤ 50 which we computed by this method.