Leonard pairs having LB-TD form

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $d\ge 3$. 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 $A,A^{\ast }$ 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 $A,A^{\ast }$ there is a nonzero scalar $q$ that is used to describe the eigenvalues of $A$ and $A^{\ast }$. In the present paper we find all Leonard pairs $A,A^{\ast }$ in ${\text{Mat}}_{d+1}(\mathbb{F})$ such that $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^{\ast }$ 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.

References in corpus (4)

• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
• Leonard pairs and the Askey-Wilson relations
• Affine transformations of a Leonard pair
• Evaluation modules for the $q$-tetrahedron algebra

Cited by in corpus (1)

• Leonard pairs having zero-diagonal TD-TD form