Primitive central idempotents of finite group rings of symmetric groups

Source abstract

Let p p be a prime. We denote by S n S_n the symmetric group of degree n n , by A n A_n the alternating group of degree n n and by F p {\mathbb F}_p the field with p p elements. An important concept of modular representation theory of a finite group G 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 {\mathbb F}_q G , where q q is a prime power. Here, we describe a new method to compute the primitive central idempotents of F q G {\mathbb F}_q G for arbitrary prime powers q q and arbitrary finite groups G G . For the group rings F p S n {\mathbb F}_p S_n of the symmetric group, we show how to derive the primitive central idempotents of F p S n − p {\mathbb F}_p S_{n-p} from the idempotents of F p S n {\mathbb F}_p S_n . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of F p S n {\mathbb F}_p S_n which contains the primitive central idempotents. The described results are most efficient for p = 2 p = 2 . In an appendix we display all primitive central idempotents of F 2 S n {\mathbb F}_2 S_n and F 4 A n {\mathbb F}_4 A_n for n ≤ 50 n \le 50 which we computed by this method.