The classification problem of P- and Q-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of P- and Q-polynomial association schemes to multivariate cases, namely to consider higher rank P- and Q-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definitions nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi [4], defined bivariate P-polynomial association schemes, as well as bivariate Q-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate P- and/or Q-polynomial association schemes.

中文翻译：

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多元 P 和/或 Q 多项式关联方案


P 和 Q 多项式关联方案的分类问题一直是代数组合学的核心问题之一。一些作者已经尝试将 P 和 Q 多项式关联方案的概念推广到多变量情况，即考虑更高等级的 P 和 Q 多项式关联方案，但到目前为止，似乎既没有非常完善的定义或结果。最近，Bernard、Crampé、d'Andecy、Vinet 和 Zaimi [4] 定义了二元 P 多项式关联方案，以及二元 Q 多项式关联方案。在本文中，我们研究了这些概念，并提出了一个关于一般单项式顺序的新修改定义，该定义更通用、更自然且易于处理。我们证明了多元 P 和/或 Q 多项式关联方案的例子有很多有趣的系列。