Modular Centralizer Algebras Corresponding to p-Groups

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We study the Loewy structure of the centralizer algebra kPQ for P a p-group with subgroup Q and k a field of characteristic p. Here kPQ is a special type of Hecke algebra. The main tool we employ is the decomposition of kPQ as a split extension of a nilpotent ideal I by the group algebra kCₚ(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 PQ. 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 kPQ.