Semisimplicity of Cellular Algebras over Positive Characteristic Fields

In this paper, we investigate semisimplicity of cellular algebras over positive characteristic fields. Our main result shows that the Frame number of cellular algebras characterizes semisimplicity of it. In a sense, this is a generalization of Maschke’s theorem. 1. intrudoction Cellular algebras are an object in algebraic combinatorics which were introduced by B. Yu. Weisfeiler and A. A. Lehman as cellular algebras and independently by D. G. Higman as coherent algebras (see [8] and [12]). They are by definition matrix algebras over a ring which is closed under the Hadamard multiplication and the transpose and containing the identity matrix and the all one matrix. Note that according to E. Bannai and T. Ito [2], a homogeneous coherent configuration is also called an association scheme (not necessarily commutative). Clearly, the adjacency algebra of a coherent configuration (or scheme) is a cellular algebra. Conversely, for each cellular algebra W there exists a coherent configuration whose adjacency algebra coincides with W . So we prefer to deal with the adjacency algebra of a coherent configuration. In a sense, cellular algebras are generalization of group algebras, so it is natural to extend Maschke’s theorem (see [10] and [11, Theorem III.1.22]) to them. Also E. Bannai and T. Ito in [3, page 303], asked about determination by the parameters, association schemes and fields for which the adjacency algebras are semisimple, symmetric, Frobenius and quasi-Frobenius. We will answer this question about semisimplicity, for general case, cellular algebras. In order to do this, we use the Frame number of cellular algebras. This number which was introduced by J. S. Frame in 1941, is in relation with the double cosets of finite groups. In 1976 D. G. Higman extended this number to cellular algebras. Z. Arad in 1999 with the help of Frame number characterized semisimplicity of commutative cellular algebras (or commutative association schemes) over fields of prime order (see [1]). Finally, A. Hanaki