The split decomposition of a tridiagonal pair

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ that satisfy (i)--(iv) below: (i) Each of $A$, $A^{\ast }$ is diagonalizable. (ii) There exists an ordering $V_0,V_1,...,V_d$ 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 $V_{-1}=0$, $V_{d+1}=0$. (iii) There exists an ordering $V_0^{\ast },V_1^{\ast },...,V_{\delta }^{\ast }$ 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 $V_{-1}^{\ast }=0$, $V_{\delta +1}^{\ast }=0$. (iv) There is no subspace $W$ of $V$ such that both $AW\subseteq W$, $A^{\ast }W\subseteq W$, other than W=0 and $W=V$. We call such a pair a tridiagonal pair on $V$. In this note we obtain two results. First, we show that each of $A,A^{\ast }$ is determined up to affine transformation by the $V_i$ and $V_i^{\ast }$. Secondly, we characterize the case in which the $V_i$ and $V_i^{\ast }$ all have dimension one. We prove both results using a certain decomposition of $V$ called the split decomposition.

7 pages

Cited by in corpus (1)

• On the shape of a tridiagonal pair