标题： 张量平方的融合强正则图 显示英文标题

标题： Fusions of the Tensor Square of a Strongly Regular Graph

摘要： 在本文中，我们确定了关联方案$\mathcal{A} \otimes \mathcal{A}$的所有融合，其中$\mathcal{A}$是对应于强正则图的对称秩$3$关联方案。 这包括所有对称秩$3$关联方案$\mathcal{A}$的保证融合，以及仅在关联方案参数存在限制时才存在的特定情况融合。 在此过程中，我们将确定强正则图的楔积的融合以及对称秩$3$表代数的张量平方的融合。 这扩展了作者和 Meagher 的近期工作，他们解决了从强正则图获得的关联方案的广义汉明方案$H(2,\mathcal{A})$的相同问题。 本文的主要结果表明 (1) 对于$\mathcal{A} \otimes \mathcal{A}$有特殊情形融合的强正则图族，与对于$H(2,\mathcal{A})$有特殊情形融合的强正则图族是同一类族；以及 (2) 不可约的强正则图是唯一一类对于楔积$\mathcal{A} \wr \mathcal{A}$有特殊情形融合的强正则图。 显示英文摘要

摘要： In this paper we determine all fusions of the association scheme $\mathcal{A} \otimes \mathcal{A}$, where $\mathcal{A}$ is the symmetric rank $3$ association scheme corresponding to a strongly regular graph. This includes both guaranteed fusions, which are fusions for all symmetric rank $3$ association schemes $\mathcal{A}$, and specific case fusions, which only exist under restrictions on the parameters of the association scheme. Along the way we will determine the fusions of wreath products of strongly regular graphs and the fusions of the tensor square of a symmetric rank $3$ table algebra. This extends recent work of the authors and Meagher, which solved the same problem for the generalized Hamming scheme $H(2,\mathcal{A})$ of the association scheme obtained from a strongly regular graph. The main results of this article show (1) the families of strongly regular graphs for which $\mathcal{A} \otimes \mathcal{A}$ has a special case fusion are the same families for which $H(2,\mathcal{A})$ has a special case fusion; and (2) the imprimitive strongly regular graphs are the only family of strongly regular graphs for which the wreath product $\mathcal{A} \wr \mathcal{A}$ has a special case fusion.