The nucleus of a $Q$-polynomial distance-regular graph

Abstract

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $\Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $\Gamma$ and the dual adjacency matrix $A^*=A^*(x)$ of $\Gamma$ with respect to $x$. We introduce a $T$-module $\mathcal N = \mathcal N(x)$ called the nucleus of $\Gamma$ with respect to $x$. We describe $\mathcal N$ from various points of view. We show that all the irreducible $T$-submodules of $\mathcal N$ are thin. Under the assumption that $\Gamma$ is a nonbipartite dual polar graph, we give an explicit basis for $\mathcal N$ and the action of $A, A^*$ on this basis. The basis is in bijection with the set of elements for the projective geometry $L_D(q)$, where $GF(q)$ is the finite field used to define $\Gamma$.