Modular adjacency algebras of Grassmann graphs

The adjacency algebra of an association scheme is defined over an arbitrary field. In general, it is always semisimple over a field of characteristic zero but not always semisimple over a field of positive characteristic. The structures of adjacency algebras over fields of positive characteristic have not been sufficiently studied.In this paper, we consider the structures of adjacency algebras of some P-polynomial schemes of class d with intersection numbers ci≢0 modulo p for 1≤i≤d over fields of positive characteristic p. The classes of these P-polynomial schemes include association schemes originating from Grassmann graphs, double Grassmann graphs, and all types of dual polar graphs. We discuss the structures of the modular adjacency algebras of Grassmann graphs.