A diagram associated with the subconstituent algebra of a distance-regular graph

Abstract

In this paper we consider a distance-regular graph $\Gamma$. Fix a vertex $x$ of $\Gamma$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\mathbb{C}$-algebra generated by the Bose-Mesner algebra $M$ of $\Gamma$ and the dual Bose-Mesner algebra $M^{\ast }$ of $\Gamma$ with respect to $x$. We consider the subspaces $M,M^{\ast },M{M}^{\ast },M^{\ast }M,M{M}^{\ast }M,M^{\ast }M{M}^{\ast },\dots$ along with their intersections and sums. In our notation, $M{M}^{\ast }$ means $Span\{RS|R\in M,S\in M^{\ast }\}$, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to $M{M}^{\ast }+M^{\ast }M$. For each subspace $U$ shown in this part of the diagram, we display an orthogonal basis for $U$ along with the dimension of $U$. For an edge $U\subseteq W$ from this part of the diagram, we display an orthogonal basis for the orthogonal complement of $U$ in $W$ along with the dimension of this orthogonal complement.