Towards a classification of the tridiagonal pairs

Kazumasa Nomura; Paul Terwilliger

paper

Towards a classification of the tridiagonal pairs

arXiv:0801.0621

Abstract

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $E n d (V)$ denote the $K$ -algebra consisting of all $K$ -linear transformations from $V$ to $V$ . We consider a pair $A, A^{*} \in E n d (V)$ that satisfy (i)--(iv) below: (i) Each of $A, A^{*}$ is diagonalizable. (ii) There exists an ordering ${V_{i}}_{i = 0}^{d}$ 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 ${V_{i}^{*}}_{i = 0}^{δ}$ of the eigenspaces of $A^{*}$ such that $A V_{i}^{*} \subseteq V_{i - 1}^{*} + V_{i}^{*} + V_{i + 1}^{*}$ for $0 \leq i \leq δ$ , where $V_{- 1}^{*} = 0$ and $V_{δ + 1}^{*} = 0$ . (iv) There is no subspace $W$ of $V$ such that $A W \subseteq W$ , $A^{*} W \subseteq W$ , $W \neq = 0$ , $W \neq = V$ . We call such a pair a {\em tridiagonal pair} on $V$ . Let $E_{0}^{*}$ denote the element of $E n d (V)$ such that $(E_{0}^{*} - I) V_{0}^{*} = 0$ and $E_{0}^{*} V_{i}^{*} = 0$ for $1 \leq i \leq d$ . Let $D$ (resp. $D^{*}$ ) denote the $K$ -subalgebra of $E n d (V)$ generated by $A$ (resp. $A^{*}$ ). In this paper we prove that the span of $E_{0}^{*} D D^{*} D E_{0}^{*}$ equals the span of $E_{0}^{*} D E_{0}^{*} D E_{0}^{*}$ , and that the elements of $E_{0}^{*} D E_{0}^{*}$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.

18 pages

Towards a classification of the tridiagonal pairs

Abstract

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