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.
Categories: math.CO