The Terwilliger algebra of the incidence graph of the Hamming graph

Abstract

Let $\varGamma$ be a distance-semiregular graph on Y, and let $D^Y$ be the diameter of $\varGamma$ on Y. Let $\varDelta$ be the halved graph of $\varGamma$ on Y. Fix $x \in Y$. Let T and $T'$ be the Terwilliger algebras of $\varGamma$ and $\varDelta$ with respect to x, respectively. Assume, for an integer i with $1 \le 2i \le D^Y$ and for $y,z \in \varGamma _{2i}(x)$ with $\partial _{\varGamma }(y,z)=2$, the numbers $|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|$ and $|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|$ depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of $\varGamma$ and $T'$-modules of $\varDelta$. Assume $\varGamma$ is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set ${\mathcal {C}}$ of all maximal cliques. Then, $\varGamma$ satisfies above assumption and $\varDelta$ is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of $\varGamma$. For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.