The search for small association schemes with noncyclotomic eigenvalues
Keywords:association schemes, table algebras, character tables
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 set 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. The main results of this paper show the eigenvalues of association schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclotomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner 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.
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