$P$-polynomial coherent configurations

Abstract

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.