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

Abstract

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\ge 1$. 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 $A^{\ast }=A^{\ast }(x)$ 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 $A,A^{\ast }$ 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 $\Gamma$.

37 pages, a few typos corrected

References in corpus (1)

• Projective geometries, $Q$-polynomial structures, and quantum groups