A group commutator involving the last distance matrix and dual distance matrix of a $Q$-polynomial distance-regular graph

Abstract: Let $\Gamma$ denote the Hamming graph $H(D,r)$ with $r \geq 3$. Consider the distance matrices ${A_i}{i=0}^{D}$ of $\Gamma$. Fix a vertex $x$ of $\Gamma$, and consider the dual distance matrices ${A_i^{*}}{i=0}^{D}$ of $\Gamma$ with respect to $x$. We investigate the group commutator $A_{D}^{-1}A_{D}^{-1}A_{D}A_{D}^{}$. We show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let $T$ denote the subconstituent algebra of $\Gamma$ with respect to $x$. We describe the action of $A_{D}^{-1}A_{D}^{-1}A_{D}A_{D}^{}$ on each irreducible $T$-module.