On closed subsets generated by a regular relation of an association scheme
On closed subsets generated by a regular relation of an association scheme
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关于由关联方案的正则关系生成的闭子集
DOI：
10.1016/j.disc.2019.111706
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发表时间：
2020
影响因子：
0.8
通讯作者：
Yoshikawa Masayoshi
中科院分区：
文献类型：
--
作者：
Yasushi Mizusawa;生田卓也;Yoshikawa Masayoshi
一个关联方案的关系s称为正则的，如果s ≠ s={s}。在Yoshikawa（2015）中，证明了对于结合方案S的每个正则关系s，s s是S的闭子集。本文证明了对于结合概型S的每一个正则关系s，< s>scinscins是一个循环群，其阶与s的强围长一致.作为这个结果的直接应用，我们得到结合概型S的正则关系的强围长整除S的阶。作为结合方案特征标理论的应用，我们得到了伯恩赛德消失定理对正则结合方案的推广，即对所有关系都是正则的结合方案的推广.我们的定理说，重数大于1的正则结合概型S的每个不可约复特征标在S的至少一个关系上为零。
A relation s of an association scheme is called regular if s∗ s s={s}. In Yoshikawa (2015), it was shown that, for each regular relation s of an association scheme S, s∗ s is a closed subset of S. In the present article, we show that, for each regular relation s of an association scheme S,< s>∕∕ s∗ s is a cyclic group the order of which coincides with the strong girth of s. As an immediate application of this result we obtain that the strong girth of a regular relation of an association scheme S divides the order of S. As an application to the theory of characters of association schemes we obtain a generalization of Burnside’s vanishing theorem to regular association schemes, that is to association schemes all relations of which are regular. Our theorem says that each irreducible complex character of a regular association scheme S with multiplicity greater than 1 vanishes on at least one relation of S.