Leonard pairs having specified end-entries

TL;DR

This paper characterizes Leonard systems of diameter d over an algebraically closed field, showing they are uniquely determined by their end-entries and a parameter beta, with specific conditions on beta and the field's characteristic.

Contribution

It provides a complete classification of Leonard systems based on end-entries and a key parameter, extending understanding of their structure and isomorphism conditions.

Findings

Leonard systems are determined by end-entries and beta under certain conditions.

Explicit conditions involving beta and field characteristic for uniqueness.

The parameter beta relates to the eigenvalue sequences and influences system classification.

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\{v_i\}_{i=0}^d$ (resp.\ $\{v^*_i\}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : V \to V$ in a similar way. The sequence $\Phi =(A, \{E_i\}_{i=0}^d, A^*, \{E^*_i\}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis $\{v_i\}_{i=0}^d$, let $\{\th_i\}_{i=0}^d$ (resp.\ $\{a^*_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp.\ $A^*$). With respect to the basis $\{v^*_i\}_{i=0}^d$, let $\{\theta^*_i\}_{i=0}^d$ (resp.\ $\{a_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A^*$ (resp.\ $A$). It is known that $\{\theta_i\}_{i=0}^d$ (resp. $\{\th^*_i\}_{i=0}^d$) are mutually distinct, and the expressions $(\theta_{i-1}-\theta_{i+2})/(\theta_i-\theta_{i+1})$, $(\theta^*_{i-1}-\theta^*_{i+2})/(\theta^*_i - \theta^*_{i+1})$ are equal and independent of $i$ for $1 \leq i \leq d-2$. Write this common value as $\beta + 1$. In the present paper we consider the "end-entries" $\theta_0$, $\theta_d$, $\theta^*_0$, $\theta^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $\beta$ if and only if either (i) $\beta \neq \pm 2$ and $q^{d-1} \neq -1$, where $\beta=q+q^{-1}$, or (ii) $\beta = \pm 2$ and $\text{Char}(\mathbb{F}) \neq 2$.