Leonard pairs from 24 points of view

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^{\ast }$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ is diagonal and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonard pair, we investigate 24 bases for $V$ on which the action of $A$ and $A^{\ast }$ takes an attractive form. With respect to each of these bases, the matrices representing $A$ and $A^{\ast }$ are either diagonal, lower bidiagonal, upper bidiagonal, or tridiagonal.

50 pages, 1 figure