On few-class Q-polynomial association schemes: feasible parameters and nonexistence results

Gavrilyuk, Alexander L.; Vidali, Janoš; Williford, Jason

doi:10.26493/1855-3974.2101.b76

Cited by 4 publications

(2 citation statements)

References 29 publications

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“…We are aware of one more feasibility condition for symmetric association schemes, the forbidden quadruple condition described in [7,Corollary 4.2]. Our 4-point transitive examples do not have any nontrivial Krein parameters equal to zero, so they satisfy this condition vacuously.…”

Section: Accepted Manuscriptmentioning

confidence: 99%

The search for small association schemes with noncyclotomic eigenvalues

Herman

Roghayeh

2023

Ars Math. Contemp.

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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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Section: Accepted Manuscriptmentioning

confidence: 99%

The search for small association schemes with noncyclotomic eigenvalues

Herman

Roghayeh

2023

Ars Math. Contemp.

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“…Moreover, this scheme is Qpolynomial (see [2] for the definitions and more results about P, Q-polynomial association schemes). Williford [8,9] compiled the tables of feasible parameters of primitive 3-class Qpolynomial association schemes (on up to 2800 vertices), where the uniqueness of the scheme W was left blank (see also [6,Appendix B]). In this note, we show that W is unique, i.e., it is determined up to isomorphism by its parameters.…”

Section: Introductionmentioning

confidence: 99%