Raising and lowering maps for tridiagonal pairs

TL;DR

This paper studies the structure of tridiagonal pairs on finite-dimensional vector spaces, focusing on the relationships among raising/lowering maps and their associated operators, and explores properties like injectivity and surjectivity.

Contribution

It provides a detailed description of the relationships among the raising/lowering maps and their associated operators in the context of tridiagonal pairs.

Findings

Relations among R, F, L, b5R, b5L are characterized

Results on injectivity and surjectivity of these maps are presented

Structural properties of the operators are analyzed

Abstract

Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A, A^*$ of maps in ${\rm End}(V)$ such that (i) each of $A, A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$ $(0 \leq i \leq d)$, where $V_{-1} =0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i \rbrace_{i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$ $(0 \leq i \leq \delta)$, where $V^*_{-1} =0$ and $V^*_{\delta+1}=0$; (iv) there does not exist a subspace $W \subseteq V$ such that $W \not=0$, $W\not=V$, $A W \subseteq W$, $A^*W \subseteq W$. Assume that $A, A^*$ is a tridiagonal pair on $V$. It is known that $d=\delta$. For $0 \leq i \leq d$ let $\theta_i$ (resp. $\theta^*_i$) denote the eigenvalue of $A$ (resp. $A^*$) for $V_i$ (resp. $V^*_i$). By construction, there exist $R,F,L \in {\rm End}(V)$ such that $A=R+F+L$ and $R V^*_i \subseteq V^*_{i+1}$, $F V^*_i \subseteq V^*_i$, $LV^*_i \subseteq V^*_{i-1}$ $(0 \leq i \leq d)$. For $0 \leq i \leq d$ define $U_i = (V^*_0 + V^*_1 + \cdots + V^*_i ) \cap (V_i + V_{i+1} + \cdots + V_d)$. It is known that the sum $V=\sum_{i=0}^d U_i$ is direct. By construction, there exists $\mathcal R, \mathcal L \in {\rm End}(V)$ such that $\mathcal R=A - \theta_i I$ and $\mathcal L= A^*-\theta^*_i I$ on $U_i$ $(0 \leq i \leq d)$. It is known that $\mathcal R U_i \subseteq U_{i+1}$ and $\mathcal L U_i \subseteq U_{i-1}$ $(0 \leq i \leq d)$, where $U_{-1}=0$ and $U_{d+1}=0$. In this paper, our main goal is to describe how $R,F,L,\mathcal R, \mathcal L$ are related. We also give some results concerning injectivity/surjectivity and $R, L$.