Title: The results of a search for small association schemes with noncyclotomic eigenvalues
|Affiliation:||University of Regina|
|Zoom:||Contact Soffia Arnadottir|
One of the open questions in Bannai and Ito's book on association schemes is the question of whether every eigenvalue of the usual adjacency matrix of every relation appearing in a commutative association scheme lies in a cyclotomic number field. They attribute this question to Simon Norton, and date it back to an Oberwolfach conference in 1980.
Forty-one years later, this question is still open, but recent evidence has been building towards it being resolved in the negative. Following the classification of association schemes of prime order, Hanaki's research team produced parameter sets for noncyclotomic symmetric association schemes of rank 5 and 6 with order in the thousands. Inspired by these examples, we set out to perform a systematic search for the smallest examples of feasible parameter sets for association schemes with noncyclotomic eigenvalues. Our approach is to first relate the association schemes with noncyclotomic eigenvalues of a given rank, involution type, and integral multiplicities to suitable integral points on an algebraic variety. We then use a computer to search for the smallest suitable points on the variety. Using this approach we have shown all association schemes of rank 4 (or less) and all asymmetric association schemes of rank 5 will have cyclotomic eigenvalues. In the case of symmetric association schemes of rank 5 we have found sixteen examples of feasible parameter sets of order less than 1000. In this talk, we will discuss our process, interesting features of these examples, and the current state of both feasibility and realizability checking for parameter sets of symmetric association schemes.
This is joint work with my PhD student Roghayeh Maleki.