Leonard pairs having LB-TD form

TL;DR

This paper classifies a specific class of Leonard pairs with LB-TD form, where one matrix is lower bidiagonal with ones on the subdiagonal and the other is irreducible tridiagonal, under certain conditions.

Contribution

It provides a complete classification of Leonard pairs with LB-TD form under the assumption that the scalar q is not a root of unity.

Findings

Identifies all Leonard pairs with LB-TD form under the given conditions.

Provides a partial solution to a problem posed by Paul Terwilliger.

Clarifies the structure of such Leonard pairs in algebraically closed fields.

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $\text{Mat}_{d+1}(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices $A,A^*$ in $\text{Mat}_{d+1}(\mathbb{F})$, each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{d+1}(\mathbb{F})$. For a Leonard pair $A,A^*$ there is a nonzero scalar $q$ that is used to describe the eigenvalues of $A$ and $A^*$. In the present paper we find all Leonard pairs $A,A^*$ in $\text{Mat}_{d+1}(\mathbb{F})$ such that $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal, under the assumption that $q$ is not a root of unity. This gives a partial solution of a problem given by Paul Terwilliger.