Leonard triples and hypercubes

TL;DR

This paper studies Leonard triples arising from the algebraic structure of hypercubes, showing that certain matrices act as Leonard triples on irreducible modules, providing detailed descriptions of these triples.

Contribution

It introduces a new connection between Leonard triples and the algebraic structure of hypercubes, specifically analyzing the Terwilliger algebra and its modules.

Findings

Leonard triples act on each irreducible module of the Terwilliger algebra of hypercubes.

Detailed characterization of Leonard triples associated with hypercube graphs.

Establishment of algebraic relations among matrices related to 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.