Primitive central idempotents of finite group rings of symmetric groups

Let p be a prime. We denote by Sn the symmetric group of degree n, by An the alternating group of degree n and by Fp 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 FqG, where q is a prime power. Here, we describe a new method to compute the primitive central idempotents of FqG for arbitrary prime powers q and arbitrary finite groups G. For the group rings FpSn of the symmetric group, we show how to derive the primitive central idempotents of FpSn−p from the idempotents of FpSn. Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of FpSn 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 F2Sn and F4An for n ≤ 50 which we computed by this method. Introduction and notation Let p be a prime, let q = p for some s ∈ N and let G be a finite group. For the finite field with q elements we write Fq, and FqG denotes the group ring of G over Fq. We use Sn and An for the symmetric and alternating group of degree n, respectively. We write Zm for a cyclic group of order m, and Zm Si for the wreath product of this group with Si. We use CG(g) for the centralizer of g ∈ G, and CG(U) for the centralizer of U ⊆ G. The centre of a group ring FG is denoted by Z(FG) and the radical of the centre by Rad(Z(FG)). We use IrrG for the set of irreducible characters of G defined over the field C of complex numbers and we write ord(g) for the order of an element g ∈ G. The symbols Z and N denote the integers and the natural numbers, respectively. A big part of modular representation theory deals with blocks. There are several possibilities to characterize blocks, for instance by the block idempotents, i.e. the primitive central idempotents of FqG. Therefore it is important to find methods to compute the primitive central idempotents. The usual method to compute primitive central idempotents is described in [8], Lemma 16.6. For this method it is necessary to compute the character table of G over the field C of complex numbers first. For the symmetric group S50 it is known that there are 204226 characters over C, but there are only 5 primitive central idempotents of F2S50. Due to the vast amount of data it is not possible to compute Received by the editor December 15, 2006 and, in revised form, March 8, 2007. 2000 Mathematics Subject Classification. Primary 20C05, 20C30, 20C40.