The end-parameters of a Leonard pair

TL;DR

This paper investigates the end-parameters of Leonard systems, showing they, along with a scalar beta, determine the system up to isomorphism and establishing an optimal upper bound on the number of systems with given end-parameters.

Contribution

It introduces the concept of end-parameters for Leonard systems and proves they, together with beta, uniquely determine the system up to isomorphism.

Findings

Leonard systems are determined by end-parameters and beta.

A relation between end-parameters and beta is established.

There are at most  systems with given end-parameters, and this bound is optimal.

Abstract

Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a 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. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $(\{\th_i\}_{i=0}^d, \{\th^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d, \{\phi_i\}_{i=1}^d)$, called its parameter array. The scalars $\{\th_i\}_{i=0}^d$ (resp.\ $\{\th^*_i\}_{i=0}^d$) are mutually distinct, and the expressions $(\th_{i-2} - \th_{i+1})/(\th_{i-1}-\th_{i})$, $(\th^*_{i-2} - \th^*_{i+1})/(\th^*_{i-1}-\th^*_{i})$ are equal and independent of $i$ for $2 \leq i \leq d-1$. Write this common value as $\beta+1$. In the present paper, we consider the "end-parameters" $\th_0$, $\th_d$, $\th^*_0$, $\th^*_d$, $\vphi_1$, $\vphi_d$, $\phi_1$, $\phi_d$ of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and $\beta$. We display a relation between the end-parameters and $\beta$. Using this relation, we show that there are up to inverse at most $\lfloor (d-1)/2 \rfloor$ Leonard systems that have specified end-parameters. The upper bound $\lfloor (d-1)/2 \rfloor$ is best possible.