A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

Abstract

Let $$\varGamma $$ denote the Hamming graph H(D, r) with $$r \ge 3$$ . Consider the distance matrices $$\{A_i\}_{i=0}^{D}$$ of $$\varGamma $$ . Fix a vertex x of $$\varGamma $$ , and consider the dual distance matrices $$\{A_i^{*}\}_{i=0}^{D}$$ of $$\varGamma $$ 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 $$\varGamma $$ with respect to x. We describe the action of $$A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$$ on each irreducible T-module.