A characterization of Leonard pairs using the notion of a tail

TL;DR

This paper characterizes Leonard pairs, special pairs of linear transformations, using the concept of a tail from algebraic graph theory, providing new insights into their structure.

Contribution

It introduces a novel characterization of Leonard pairs through the notion of a tail, bridging algebraic graph theory and linear algebra.

Findings

Provides a new characterization of Leonard pairs

Connects algebraic graph theory with linear algebra concepts

Enhances understanding of the structure of Leonard pairs

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ 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 $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.