Association schemes and permutation groups

Every permutation group which is not 2-transitive acts on a nontrivial coherent configuration, but the question of which permutation groups G act on nontrivial association schemes (symmetric coherent configurations) is considerably more subtle. A closely related question is: when is there a unique minimal G-invariant association scheme? We examine these questions, and relate them to more familiar concepts of permutation group theory (such as generous transitivity) and association scheme theory (such as stratifiability). Our main results are the determination of all regular groups having a unique minimal association scheme, and a classification of groups with no non-trivial association scheme. The latter must be primitive, and are either 2-homogeneous, almost simple, or of diagonal type. The diagonal groups have some very interesting features, and we examine them further. Among other things we show that a diagonal group with non-abelian base group cannot be stratifiable if it has ten or more factors, or generously transitive if it has nine or more; and we characterise the quaternion group Q8 as the unique non-abelian group T such that a diagonal group with eight factors T is generously transitive. 1 Association schemes and coherent configurations A coherent algebra, or cellular algebra, is an algebra of n× n complex matrices which has a basis {B0,B1, . . . ,Bt} consisting of matrices with entries 0 and 1 satisfying the following conditions: (a) B0 + B1 + · · ·+ Bt = J, where J is the all-1 matrix; 1 Corresponding author 2 This research was conducted while the first author was visiting Queen Mary, University of London. The financial support of the University of the Philippines Academic Distinction Awards is gratefully acknowledged. Preprint submitted to Elsevier Preprint 28 June 2001 (b) there is a subset of {B0, . . . ,Bt} with sum I, the identity matrix; (c) the set {B0, . . . ,Bt} is closed under transposition. Since these matrices span an algebra, we have (d) BiB j = t ∑ k=0 bi jBk, where the bi j are complex numbers. The algebra is called homogeneous if the subset referred to in (b) contains just one element, which we take to be B0 = I. Any n×n zero-one matrix can be regarded as the characteristic function of a subset of Ω×Ω, where Ω = {1, . . . ,n}. Condition (a) says that the matrices B0, . . . ,Bt correspond to a partition of Ω2, and the other conditions can be translated into combinatorial statements about this partition. Thus, if the parts are C0, . . . ,Ct , then a subset of these sets partitions the diagonal; the transpose (or converse) of each part is another part; and, if (x,y) ∈Ck, then the number of points z for which (x,z) ∈Ci and (z,y) ∈C j is equal to bi j, independent of the choice of (x,y). (So, incidentally, we see that the numbers bi j are non-negative integers.) Such a combinatorial object is called a coherent configuration. Conversely, any coherent configuration gives rise to a coherent algebra. Given any permutation group G on Ω, we obtain a coherent configuration C(G) whose classes are the orbits of G on Ω2. Its coherent algebra, which we will denote by K(G), is the centraliser algebra of G, the algebra of all matrices commuting with the permutation matrices arising from elements of G. The dimension of K(G) is equal to the number of orbits of G on Ω2: this number is called the rank of the permutation group G. Note that K(G) is homogeneous if and only if G is transitive, and K(G) has the smallest possible dimension (namely 2) if and only if G is 2-transitive. An association scheme is a coherent configuration in which each of the sets Ci is self-converse (closed under reversing the order of the elements in each pair). It is easy to show that an association scheme is homogeneous, so that B0 = I. The classes C1, . . . ,Ct can now be identified with sets of unordered pairs (2-element subsets of Ω), which partition the set of all 2-element subsets of Ω. The coherent algebra of an association scheme is called its Bose–Mesner algebra. Since all its matrices are symmetric, it is commutative. A trivial example of a Bose–Mesner algebra is spanned by the two matrices I and J− I. Any other such algebra or association scheme is called non-trivial. Note that C(G) is trivial if and only if G is 2-transitive.