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 $A_n$ the alternating group of degree $n$ and by ${\mathbb F}_p$ 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 ${\mathbb F}_q G$, where $q$ is a prime power. Here, we describe a new method to compute the primitive central idempotents of ${\mathbb F}_q G$ for arbitrary prime powers $q$ and arbitrary finite groups $G$. For the group rings ${\mathbb F}_p S_n$ of the symmetric group, we show how to derive the primitive central idempotents of ${\mathbb F}_p S_{n-p}$ from the idempotents of ${\mathbb F}_p S_n$. Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of ${\mathbb F}_p S_n$ 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 ${\mathbb F}_2 S_n$ and ${\mathbb F}_4 A_n$ for $n \le 50$ which we computed by this method.