Cyclotomic coset association schemes

Abstract

For any abelian group G and subgroup A of Aut(G), let {Ci} be the A-orbits in G. The cyclotomic coset scheme is the commutative, non-symmetric association scheme with point set G and ith relation {(x, y) : xy–1 is in Ci}. This scheme admits the semidirect product of G by A as a group of automorphisms and is a generalization of cyclotomic association schemes. We develop the theory of the cyclotomic coset scheme with particular attention to its character table. We give constructions for the character table and a formula for its inverse. If A contains the pth power map for a prime p not dividing |G| then the character table is integral modulo powers of prime ideals π over (p). The mod πt character table is used to construct all factors modulo pt of scheme elements. We use the mod π character table to link the set-ness property of scheme elements to their character values. We show that the techniques for the study of difference sets and related structures persist in this new context. Cyclotomic coset schemes provide new tools to exploit the faithful spectrum. We apply this theory to a conjecture of John Dillon on cyclic difference sets with classical parameters. We also compute and use faithful idempotents modulo 2 to prove that a counter example to Dillon's conjecture must be a mod 2 sum of cosets of nontrivial subgroups of G.