Coherent configurations derived from quasiregular points in generalized quadrangles

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“…Payne [58] found that the second subconstituent of the collinearity graph of a generalized quadrangle with respect to a quasiregular point is a 3-class association scheme (or a strongly regular graph). Together with Hobart [45] he found conditions to embed the association scheme back in a generalized quadrangle.…”

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“…With fixed x and a > 1 (necessarily a is odd), the numbers of points y with k = 0 and with k = q + 1 are positive and easily computed. Payne observed in [4] that the points at distance 2 from such a quasiregular point p form an association scheme A(S,p) with three classes with one of the relations being collinearity.…”

“…In this note, we consider the analogous question for the quadrangle association schemes of [4]. The techniques used are mostly eigenvalue techniques, and [2] is a general reference for these.…”

“…Throughout, we will assume that (X, {f 1 ,f 2 ,f 3 }) is an association scheme with parameters as in [4] such that all maximal cliques in f l = (X, f 1 ) have size s. We will call such maximal cliques lines, the elements of X points, and refer to x and y as adjacent if (x,y) e f 1 . Let A i be the adjacency matrix and M i be the intersection matrix for the relation f i ; so A i is the matrix with rows and columns indexed by elements of X with (x, y) entry equal to 1 if (x, y) e f i and 0 otherwise, and M i is the 4 x 4 matrix with (j, k) entry p k ji .…”

“…Our notation differs from [4] in that we use only the parameters q and a; the remaining parameter t satisfies t = qs. Our main result is:…”