The Terwilliger Algebra of a Distance-Regular Graph of Negative Type

TL;DR

This paper investigates the structure of the Terwilliger algebra associated with a specific class of distance-regular graphs of negative type, revealing the existence and properties of certain irreducible modules.

Contribution

It characterizes the irreducible modules of the Terwilliger algebra for graphs with classical parameters and negative type, providing explicit bases and actions.

Findings

Exactly two irreducible T-modules with endpoint 1 exist up to isomorphism.

The dimensions of these modules are D and 2D-2.

Explicit bases and actions of A on these bases are provided.

Abstract

Let $\Gamma$ denote a distance-regular graph with diameter $D \ge 3$. Assume $\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\Gamma$ and let $A \in MX$ denote the adjacency matrix of $\Gamma$. Fix $x \in X$ and let $A^* \in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^*$. We call $T$ the {\em Terwilliger algebra} of $\Gamma$ with respect to $x$. We show that up to isomorphism there exist exactly two irreducible $T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these $T$-modules we display a basis consisting of eigenvectors for $A^*$, and for each basis we give the action of $A$