Raising and lowering maps for tridiagonal pairs

Abstract

Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A,A^{\ast }$ of maps in $\mathrm{End}(V)$ such that (i) each of $A,A^{\ast }$ is diagonalizable; (ii) there exists an ordering $\{V_i{\}}_{i=0}^d$ of the eigenspaces of $A$ such that $A^{\ast }{V}_i\subseteq V_{i-1}+V_i+V_{i+1}$ $(0\le i\le d)$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\{V_i^{\ast }{\}}_{i=0}^{\hat{I}}´$ of the eigenspaces of $A^{\ast }$ such that $A{V}_i^{\ast }\subseteq V_{i-1}^{\ast }+V_i^{\ast }+V_{i+1}^{\ast }$ $(0\le i\le \hat{I}´)$, where $V_{-1}^{\ast }=0$ and $V_{\hat{I}´+1}^{\ast }=0$; (iv) there does not exist a subspace $W\subseteq V$ such that $W\ne 0$, $W\ne V$, $AW\subseteq W$, $A^{\ast }W\subseteq W$. Assume that $A,A^{\ast }$ is a tridiagonal pair on $V$. It is known that $d=\hat{I}´$. For $0\le i\le d$ let $\hat{I}{¸}_i$ (resp. $\hat{I}{¸}_i^{\ast }$) denote the eigenvalue of $A$ (resp. $A^{\ast }$) for $V_i$ (resp. $V_i^{\ast }$). By construction, there exist $R,F,L\in \mathrm{End}(V)$ such that $A=R+F+L$ and $R{V}_i^{\ast }\subseteq V_{i+1}^{\ast }$, $F{V}_i^{\ast }\subseteq V_i^{\ast }$, $L{V}_i^{\ast }\subseteq V_{i-1}^{\ast }$ $(0\le i\le d)$. For $0\le i\le d$ define $U_i=(V_0^{\ast }+V_1^{\ast }+\cdots +V_i^{\ast })\cap (V_i+V_{i+1}+\cdots +V_d)$. It is known that the sum $V={\sum }_{i=0}^d{U}_i$ is direct. By construction, there exists $\mathcal{R},\mathcal{L}\in \mathrm{End}(V)$ such that $\mathcal{R}=A-\hat{I}{¸}_{i}I$ and $\mathcal{L}=A^{\ast }-\hat{I}{¸}_i^{\ast }I$ on $U_i$ $(0\le i\le d)$. It is known that $\mathcal{R}{U}_i\subseteq U_{i+1}$ and $\mathcal{L}{U}_i\subseteq U_{i-1}$ $(0\le i\le d)$, 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 $R,L$.

30 pages