The classification of tridiagonal pairs

Paul Terwilliger (University of Wisconsin - Madison, USA)

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 $A^*:V \to V$ that satisfy the following conditions:

(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 $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, 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 $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1}=0$ and $V^*_{\delta+1}=0$;

(iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$.

We call such a pair a tridiagonal pair on $V$. We classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We discuss the connection to $Q$-polynomial distance-regular graphs and the orthogonal polynomials from the terminating branch of the Askey-scheme. This is joint work with Tatsuro Ito and Kazumasa Nomura.