Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes

TL;DR

This paper proves that certain complex mathematical structures called imprimitive Q-polynomial association schemes with six classes do not exist, completing a classification that aligns with known results for related graph structures.

Contribution

It establishes the nonexistence of imprimitive Q-polynomial association schemes with six classes, filling a gap in the classification of these mathematical objects.

Findings

Proves nonexistence of six-class imprimitive Q-polynomial association schemes.

Completes the classification aligning with known results for distance-regular graphs.

Supports Suzuki's theorem by ruling out the last remaining case.

Abstract

Suzuki (1998) showed that an imprimitive Q-polynomial association scheme with first multiplicity at least three is either Q-bipartite, Q-antipodal, or with four or six classes. The exceptional case with four classes has recently been ruled out by Cerzo and Suzuki (2009). In this paper, we show the nonexistence of the last case with six classes. Hence Suzuki's theorem now exactly mirrors its well-known counterpart for imprimitive distance-regular graphs.