The search for small association schemes with noncyclotomic eigenvalues

TL;DR

This paper investigates the smallest possible parameters for commutative association schemes with noncyclotomic eigenvalues, showing that certain low-rank schemes must have cyclotomic eigenvalues, with some exceptions in rank 5.

Contribution

It determines feasible parameter sets for small association schemes with noncyclotomic eigenvalues and proves that schemes of rank 4 and certain rank 5 schemes are necessarily cyclotomic.

Findings

Rank 4 commutative association schemes have cyclotomic eigenvalues.

Rank 5 nonsymmetric schemes also have cyclotomic eigenvalues.

Examples of rank 5 schemes with noncyclotomic eigenvalues are provided.

Abstract

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic variety for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. Our main results show the eigenvalues of commutative association schemes of rank 4 and nonsymmetric commutative association schemes of rank 5 will always be cyclotomic. In the rank 5 cases these our conclusions rely on calculations done by computer for Gr\"obner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.