The cyclotomic eigenvalue and character value problems for association schemes

Abstract

The central focus of this dissertation is an answer to the cyclotomic eigenvalue question (CEQ) for commutative association schemes. This question, which was posed by Simon Norton at Oberwolfach in 1980, asks whether all entries of the character table of a commutative association scheme lie in a cyclotomic extension of the rational numbers. In this thesis, we consider the cyclotomic eigenvalue question for objects that generalize the notion of association schemes. The objects in question are standard integral table algebras with integral multiplicities (SITAwIMs). Our main results show the eigenvalues of SITAwIMs of rank 4 and nonsymmetric ones of rank 5 are cyclotomic. This implies that the CEQ is affirmative for association schemes of rank 4 and nonsymmetric association schemes of rank 5. Moreover, for rank 5 symmetric SITAwIMs we give several examples that have noncyclotomic eigenvalues, and show the parameters for many of these examples satisfy all known parameter requirements for association schemes. Next, we provide an algorithm for computing fusions of a based algebra (A;B). We apply this algorithm to three problems: (1) computing fusion lattices for association schemes,of order up to 30; (2) realizing association schemes with transitive automorphism groups; (3) producing small examples of non-Schurian association schemes with noncyclotomic eigenvalues. We give an example of a group of order 96 for which two of its Schurian association scheme quotients have non-Schurian fusions with noncyclotomic eigenvalues. The latter is of interest to the open question asking whether association schemes with transitive automorphism groups can have noncyclotomic character values.