Affine transformations of a Leonard pair

TL;DR

This paper investigates conditions under which affine transformations of Leonard pairs preserve their structure and isomorphism classes, providing a detailed algebraic characterization of these transformations.

Contribution

It establishes necessary and sufficient conditions for affine transformations of Leonard pairs to remain isomorphic to the original or swapped pairs, advancing understanding of their symmetry properties.

Findings

Characterization of when affine transformations preserve Leonard pair structure

Conditions for isomorphism between transformed and original Leonard pairs

Analysis of symmetry and duality in Leonard pairs

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ 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 $V$. Let $x$, $c$, $x^*$, $c^*$ denote scalars in $K$ with $x$, $x^*$ nonzero, and note that $xA+cI$, $x^*A^* + c^*I$ is a Leonard pair on $V$. We give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A$, $A^*$. We also give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A^*$, $A$.