Modular Centralizer Algebras Corresponding to p-Groups

TL;DR

This paper analyzes the structure of centralizer algebras associated with p-groups, providing explicit Loewy series, bounds on Loewy length, and determining their representation type using computational methods.

Contribution

It introduces a detailed study of the Loewy structure of kP^Q algebras, including decomposition techniques, bounds, and computational approaches for various p-groups.

Findings

Computed Loewy structures for several classes of p-groups

Established bounds on Loewy length of kP^Q

Determined the representation type of the algebra

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.