Modular Centralizer Algebras Corresponding to p-Groups

Abstract

We study the Loewy structure of the centralizer algebra kP^Q for P a p-group with subgroup Q and k a field of characteristic p. Here kP^Q is a special type of Hecke algebra. The main tool we employ is the decomposition of kP^Q as a split extension of a nilpotent ideal I by the group algebra kC_P(Q). We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of $P^Q. Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As a final application of the decomposition, we determine the representation type of kP^Q.