Modular adjacency algebras, standard representations, and p-ranks of cyclotomic association schemes

摘要

In this paper, we consider cyclotomic association schemes S = Cyc(p(a), d). We focus on the adjacency algebra of S over algebraically closed fields K of characteristic p. If p equivalent to 1 (mod d), p equivalent to -1 (mod d), or d is an element of {2, 3, 4, 5, 6}, we identify the adjacency algebra of S over K as a quotient of a polynomial ring over an admissible ideal. In several cases, we determine the indecomposable direct sum decomposition of the standard module of S. As a consequence, we are able to compute the p-rank of several specific elements of the adjacency algebra of S over K.