Leonard pairs having zero-diagonal TD-TD form

TL;DR

This paper classifies all Leonard pairs of zero-diagonal irreducible tridiagonal matrices over an algebraically closed field, solving a problem posed by Paul Terwilliger.

Contribution

It provides a complete characterization of zero-diagonal Leonard pairs in matrix algebra, extending the understanding of their structure.

Findings

All such Leonard pairs are explicitly described.

The classification solves an open problem by Terwilliger.

The results deepen the understanding of Leonard pairs with special structure.

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let $\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n \times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices in $\text{Mat}_{n}(\mathbb{F})$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{n}(\mathbb{F})$. In the present paper, we find all Leonard pairs $A,A^*$ in $\text{Mat}_{n}(\mathbb{F})$ such that each of $A$ and $A^*$ is irreducible tridiagonal with all diagonal entries $0$. This solves a problem given by Paul Terwilliger.