How to recognize a Leonard pair

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy (i) and (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In the literature, there are some parameters that are used to describe Leonard pairs called the intersection numbers $\{a_i{\}}_{i=0}^d$, $\{b_i{\}}_{i=0}^{d-1}$, $\{c_i{\}}_{i=1}^d$, and the dual eigenvalues $\{{\theta }_i^{\ast }{\}}_{i=0}^d$. In this paper, we provide two characterizations of Leonard pairs. For the first characterization, the focus is on the $\{a_i{\}}_{i=0}^d$ and $\{{\theta }_i^{\ast }{\}}_{i=0}^d$. For the second characterization, the focus is on the $\{b_i{\}}_{i=0}^{d-1}$, $\{c_i{\}}_{i=1}^d$, and $\{{\theta }_i^{\ast }{\}}_{i=0}^d$.

17 pages

References in corpus (1)

• A characterization of Leonard pairs using the parameters $\{a_i{\}}_{i=0}^d$