Finite simple groups with non-determined group association schemes

Determine for which non-abelian finite simple groups the group association scheme is not determined up to combinatorial isomorphism by its intersection numbers.

Background

The paper constructs algebraically isomorphic but combinatorially non-isomorphic Schur or Cayley association schemes for an infinite family of groups PSL(2,q), as well as for the alternating groups A_6 and A_8. These examples show that primitivity does not force a group association scheme to be determined by its intersection numbers.

The authors ask for a classification across all non-abelian finite simple groups of those whose group association schemes fail this determination property. This is broader than the specific PSL(2,q), A_6, and A_8 constructions established in the paper.

References

For which non-abelian finite simple groups $G$ is the group association scheme $\mathfrak X(G)$ not determined up to combinatorial isomorphism by its intersection numbers?

— Twisted primitive group association schemes  (2608.16278 - Higashitani et al., 17 Aug 2026) in Section 6, “Summary and future questions”

For $PSL(2,q)$, $\mathfrak A_6$ and $\mathfrak A_8$, are there further twists, not equivalent to the ones constructed here, which are algebraically isomorphic but combinatorially non-isomorphic to the original group association schemes? Do there exist association schemes algebraically isomorphic to $\mathfrak{X}(PSL(2,q))$, $\mathfrak X(\mathfrak A_6)$ or $\mathfrak X(\mathfrak A_8)$ which are not Cayley association schemes?

— Twisted primitive group association schemes  (2608.16278 - Higashitani et al., 17 Aug 2026) in Section 6, “Summary and future questions”