Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple

Abstract

Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional subspaces of $V$ whose sum is $V$. For a linear transformation $A$ from $V$ to $V$, we say $A$ lowers $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i-1}$ for $0 \leq i \leq d$, where $V_{-1}=0$. We say $A$ raises $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{d+1}=0$. An ordered pair of linear transformations $A,B$ from $V$ to $V$ is called LR whenever there exists a decomposition $\{V_i\}_{i=0}^d$ of $V$ that is lowered by $A$ and raised by $B$. In this case the decomposition $\{V_i\}_{i=0}^d$ is uniquely determined by $A,B$; we call it the $(A,B)$-decomposition of $V$. Consider a $3$-tuple of linear transformations $A$, $B$, $C$ from $V$ to $V$ such that any two of $A$, $B$, $C$ form an LR pair on $V$. Such a $3$-tuple is called an LR triple on $V$. Let $α$, $β$, $γ$ be nonzero scalars in $\mathbb{F}$. The triple $αA, βB, γC$ is an LR triple on $V$, said to be associated to $A,B,C$. Let $\{V_i\}_{i=0}^d$ be a decomposition of $V$ and let $X$ be a linear transformation from $V$ to $V$. We say $X$ is tridiagonal with respect to $\{V_i\}_{i=0}^d$ whenever $X V_i \subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$. Let $\cal X$ be the vector space over $\mathbb{F}$ consisting of the linear transformations from $V$ to $V$ that are tridiagonal with respect to the $(A,B)$ and $(B,C)$ and $(C,A)$ decompositions of $V$. There is a special class of LR triples, called $q$-Weyl type. In the present paper, we find a basis of $\cal X$ for each LR triple that is not associated to an LR triple of $q$-Weyl type.