Leonard triples and hypercubes

Abstract

Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A$ denote the adjacency matrix of $Q_D$. Fix $x \in X$ and let $A^*$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $Mat_X(C)$ generated by $A, A^*$. We refer to $T$ as the {\em Terwilliger algebra of} $Q_D$ {\em with respect to} $x$. The matrices $A$ and $A^*$ are related by the fact that $2 \im A = A^* A^e - A^e A^*$ and $2 \im A^* = A^e A - A A^e$, where $2 \im A^e = A A^* - A^* A$ and $\im^2=-1$. We show that the triple $A$, $A^*$, $A^e$ acts on each irreducible $T$-module as a Leonard triple. We give a detailed description of these Leonard triples.