扭本原群结合方案
Twisted primitive group association schemes
AI总结：
本研究针对本原群结合方案的相交数是否决定其组合同构类的问题，构造了多类与群共轭类分划代数同构但组合不同构的Schur分划，证明相关本原群结合方案的相交数无法确定其组合同构类。
AI中文摘要：
我们针对本原群结合方案的相交数是否能在组合同构意义下确定该方案这一问题给出研究结果。对于$G=\operatorname{PSL}(2,q)$，其中$q$为满足$q=11$或$q\ge 17$的奇素数幂，或是满足$f\ge3$的$q=2^f$，我们构造了一种Schur分划（舒尔分划），它与$G$的共轭类分划代数同构，但组合不同构。由此可知，对应的本原群结合方案无法通过其相交数在组合同构意义下被确定；具体而言，它们是不可分的。对于$\mathfrak A_6$和$\mathfrak A_8$，我们也显式构造了与对应共轭类分划代数同构但组合不同构的Schur分划。
英文摘要：
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.
