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arXiv · 2608.16278

Twisted primitive group association schemes

Akihiro Higashitani·Masanari Kamiya·Hirotake Kurihara
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Abstract

We give results on the question of whether the intersection numbers of a primitive group association scheme determine it up to combinatorial isomorphism. For $G=\operatorname{PSL}(2,q)$, where $q$ is an odd prime power with $q=11$ or $q\ge 17$, or $q=2^f$ with $f\ge3$, we construct a Schur partition that is algebraically isomorphic to the partition of $G$ into conjugacy classes but not combinatorially isomorphic to it. Consequently, the corresponding primitive group association schemes are not determined up to combinatorial isomorphism by their intersection numbers; in particular, they are non-separable. For $\mathfrak A_6$ and $\mathfrak A_8$, we also explicitly construct Schur partitions that are algebraically isomorphic to the corresponding partitions into conjugacy classes but not combinatorially isomorphic to them.

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Akihiro Higashitani, Masanari Kamiya, Hirotake Kurihara. 2026-08-17. Twisted primitive group association schemes. https://arxiv.org/abs/2608.16278

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