The structure 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 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 tridiagonal pair on $V$. It is known that $d=\delta$ and for $0\le i\le d$ the dimensions of $V_i,V_{d-i},V_i^{\ast },V_{d-i}^{\ast }$ coincide. In this paper we show that the following (i)--(iv) hold provided that $K$ is algebraically closed: (i) Each of $V_0$, $V_0^{\ast }$, $V_d$, $V_d^{\ast }$ has dimension 1. (ii) There exists a nondegenerate symmetric bilinear form $(,)$ on $V$ such that $(Au,v)=(u,Av)$ and $(A^{\ast }u,v)=(u,A^{\ast }v)$ for all $u,v\in V$. (iii) There exists a unique anti-automorphism of $End(V)$ that fixes each of $A,A^{\ast }$. (iv) The pair $A,A^{\ast }$ is determined up to isomorphism by the data $(\{{þ}_i{\}}_{i=0}^d;\{{þ}_i^{\ast }{\}}_{i=0}^d;\{{\zeta }_i{\}}_{i=0}^d)$, where ${þ}_i$ (resp. ${þ}_i^{\ast }$) is the eigenvalue of $A$ (resp. $A^{\ast }$) on $V_i$ (resp. $V_i^{\ast }$), and $\{{\zeta }_i{\}}_{i=0}^d$ is the split sequence of $A,A^{\ast }$ corresponding to $\{{þ}_i{\}}_{i=0}^d$ and $\{{þ}_i^{\ast }{\}}_{i=0}^d$.

18 pages

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Cited by in corpus (3)

• On relative $t$-designs in polynomial association schemes
• Tridiagonal pairs and the q-tetrahedron algebra
• On the shape of a tridiagonal pair