AI 中文总结
本文针对P-多项式相干构型提出与Q-多项式框架契合的替代定义，证明拉托定义的双纤维P-多项式构型符合该定义，还给出三类多纤维P-多项式构型并验证其满足跨块交矩阵为三对角的条件。
AI 中文摘要
须田（Suda）引入了Q-多项式相干构型的概念，这是一个自然且重要的概念。随后，拉托（Lato）引入了P-多项式相干构型的概念，并证明每个满足其定义的此类构型至多有两个纤维。尽管拉托的定义很有趣，尤其是因为它刻画了双正则距离图，但我们认为需要一个替代定义。本文中，我们提出了一个与须田的Q-多项式框架自然契合的P-多项式相干构型的替代概念。我们证明，每个在拉托意义下是P-多项式的双纤维相干构型，在我们的意义下也是P-多项式的，反之则不成立。我们进一步证明，每个类型为(2,2;3)、(3,2;3)或(3,3;3)的相干构型在我们的意义下是P-多项式的。此外，我们给出了具有任意数量纤维的三类P-多项式相干构型：来自ℝ²中紧欧氏t-设计的构型、H(n,2)的Terwilliger代数对应的构型，以及𝔽_qⁿ中所有子空间构成的构型。最后，我们给出了跨块交矩阵为三对角的等价条件，并验证这三类构型均满足该条件。
英文摘要
Suda introduced the notion of a $Q$-polynomial coherent configuration, which provides a natural and important concept. Subsequently, Lato introduced a notion of a $P$-polynomial coherent configuration and proved that every such configuration satisfying the definition has at most two fibers. Although Lato's definition is interesting, particularly because it characterizes distance-biregular graphs, we argue that an alternative definition is desirable. In this paper, we propose an alternative notion of $P$-polynomial coherent configurations that is naturally aligned with Suda's $Q$-polynomial framework. We show that every two-fiber coherent configuration that is $P$-polynomial in Lato's sense is also $P$-polynomial in our sense, whereas the converse does not hold. We further prove that every coherent configuration of type $(2,2;3)$, $(3,2;3)$ or $(3,3;3)$ is $P$-polynomial in our sense. In addition, we present three families of $P$-polynomial coherent configurations with an arbitrary number of fibers: those arising from tight Euclidean $t$-designs in $\mathbb R^2$, the Terwilliger algebra of $H(n,2)$, and the set of all subspaces of $\mathbb F_q^n$. Finally, we give an equivalent condition for the cross-block intersection matrices to be tridiagonal and verify that all three families satisfy this condition.
Comments25 pages
