How to sharpen a tridiagonal pair

Abstract

Let $\F$ denote a field and let $V$ denote a vector space over $\F$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy the following conditions: (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 $A^{\ast }{V}_i\subseteq V_{i-1}+V_i+V_{i+1}$ for $0\le i\le d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\{V_i^{\ast }{\}}_{i=0}^{\delta }$ of the eigenspaces of $A^{\ast }$ such that $A{V}_i^{\ast }\subseteq V_{i-1}^{\ast }+V_i^{\ast }+V_{i+1}^{\ast }$ for $0\le i\le \delta$, where $V_{-1}^{\ast }=0$ and $V_{\delta +1}^{\ast }=0$; (iv) there is no subspace $W$ of $V$ such that $AW\subseteq W$, $A^{\ast }W\subseteq W$, $W\ne 0$, $W\ne V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=\delta$, and for $0\le i\le d$ the dimensions of $V_i,V_i^{\ast },V_{d-i},V_{d-i}^{\ast }$ coincide. Denote this common dimension by ${\rho }_i$ and call $A,A^{\ast }$ {\it sharp} whenever ${\rho }_0=1$. Let $T$ denote the $\F$-subalgebra of ${\rm End}_\F(V)$ generated by $A,A^{\ast }$. We show: (i) the center $Z(T)$ is a field whose dimension over $\F$ is ${\rho }_0$; (ii) the field $Z(T)$ is isomorphic to each of $E_{0}T{E}_0$, $E_{d}T{E}_d$, $E_0^{\ast }T{E}_0^{\ast }$, $E_d^{\ast }T{E}_d^{\ast }$, where $E_i$ (resp. $E_i^{\ast }$) is the primitive idempotent of $A$ (resp. $A^{\ast }$) associated with $V_i$ (resp. $V_i^{\ast }$); (iii) with respect to the $Z(T)$-vector space $V$ the pair $A,A^{\ast }$ is a sharp tridiagonal pair.

10 pages

References in corpus (2)

• Finite-dimensional irreducible modules for the three-point ${\mathfrak{sl}}_2$ loop algebra
• Towards a classification of the tridiagonal pairs