Generalized quadrangles and subconstituent algebra ¹

Abstract

The point graph of a generalized quadrangle GQ (s, t) is a strongly regular graph G = srg(u,kappa,lamda,mu) whose parameters depend on s and t. By a detailed analysis of the adjacency matrix we compute the Terwilliger algebra of this kind of graphs (and denoted it by T). We find that there are only two non-isomorphic Terwilliger algebras for all the generalized quadrangles. The two classes correspond to wether s^2 = t or not. We decompose the algebra into direct sum of simple ideals. Considering the action of T on C^X we find the decomposition into irreducible T-submodules of C^X (where X is the set of vertices of the graph G ).