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

TL;DR

This paper introduces a diagrammatic representation of the subspaces generated by the Bose-Mesner algebra and its dual in the subconstituent algebra of a distance-regular graph, detailing their relations and bases.

Contribution

It presents a novel diagram that visualizes the relationships among key subspaces in the subconstituent algebra of a distance-regular graph, including bases and dimensions.

Findings

Detailed diagram of subspace relations up to $MM^*+M^*M$

Orthogonal bases and dimensions for each subspace

Orthogonal complements and their bases illustrated

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 $M^*$ of $\Gamma$ with respect to $x$. We consider the subspaces $M, M^*, MM^*, M^*M, MM^*M, M^*MM^*, \dots$ along with their intersections and sums. In our notation, $MM^*$ means $Span\{RS|R\in M, S\in M^*\}$, 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 $MM^*+M^*M$. 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.