On Pseudocyclic Association Schemes

TL;DR

This paper generalizes pseudocyclic association schemes to non-commutative cases, showing that schemes with high rank relative to valency are Frobenius schemes and are uniquely determined by their intersection numbers, with implications for schemes of prime degree.

Contribution

It extends the theory of pseudocyclic schemes to non-commutative cases and characterizes high-rank schemes as Frobenius schemes, establishing their uniqueness by intersection numbers.

Findings

High-rank pseudocyclic schemes are Frobenius schemes.

Such schemes are uniquely determined by their intersection number array.

Schemes of prime degree with large enough rank are schurian.

Abstract

The notion of pseudocyclic association scheme is generalized to the non-commutative case. It is proved that any pseudocyclic scheme the rank of which is much more than the valency is the scheme of a Frobenius group and is uniquely determined up to isomorphism by its intersection number array. An immediate corollary of this result is that any scheme of prime degree, valency $k$ and rank at least $k^4$ is schurian.