Leonard pairs having zero-diagonal TD-TD form

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $n\ge 1$. 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 $A,A^{\ast }$ in ${\text{Mat}}_n(\mathbb{F})$ such that each of $A$ and $A^{\ast }$ is irreducible tridiagonal with all diagonal entries $0$. This solves a problem given by Paul Terwilliger.

References in corpus (6)

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• 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
• The classification of Leonard triples of QRacah type
• Affine transformations of a Leonard pair
• Evaluation modules for the $q$-tetrahedron algebra
• Leonard pairs having LB-TD form