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

Abstract

In this paper, we consider cyclotomic association schemes $S = \mathrm {Cyc}(p^a, d)$. We focus on the adjacency algebra of S over algebraically closed fields K of characteristic p. If $p\equiv 1 \pmod {d}$, $p\equiv -1\pmod {d}$, or $d\in \{2,3,4,5,6\}$, 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.