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

TL;DR

This paper introduces and analyzes the nucleus of a $Q$-polynomial distance-regular graph, revealing its structure, properties, and explicit bases in the case of nonbipartite dual polar graphs, linking it to projective geometry.

Contribution

It defines the nucleus of such graphs, studies its irreducible submodules, and provides explicit bases and actions for nonbipartite dual polar graphs, connecting algebraic and geometric structures.

Findings

All irreducible $T$-submodules of the nucleus are thin.

Explicit basis for the nucleus in nonbipartite dual polar graphs.

The basis corresponds to elements of the projective geometry $L_D(q)$.

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 $\Gamma$.