Leonard triples of $q$-Racah type

TL;DR

This paper characterizes Leonard triples of $q$-Racah type by constructing specific invertible endomorphisms that relate the triples' elements through conjugation, revealing their mutual commutation properties.

Contribution

It establishes the existence and uniqueness of certain invertible elements that relate the components of $q$-Racah type Leonard triples, and explores their commutation relations.

Findings

Existence of invertible elements W, W', W'' with specific conjugation properties.

Mutual commutation of products W'W, W''W', and WW'','

Product of these elements is a scalar multiple of the identity.

Abstract

Let $\mathbb F$ denote a field, and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. Pick a nonzero $q \in \mathbb F$ such that $q^4 \not=1$, and let $A,B,C$ denote a Leonard triple on $V$ that has $q$-Racah type. We show that there exist invertible $W, W', W'' $ in ${\rm End}(V)$ such that (i) $A$ commutes with $W$ and $W^{-1}BW-C$; (ii) $B$ commutes with $W'$ and $(W')^{-1}CW'-A$; (iii) $C$ commutes with $W''$ and $(W'')^{-1}AW''-B$. Moreover each of $W,W', W''$ is unique up to multiplication by a nonzero scalar in $\mathbb F$. We show that the three elements $W'W, W''W', WW''$ mutually commute, and their product is a scalar multiple of the identity. A number of related results are obtained.