Tridiagonal pairs of $q$-Racah type and the $μ$-conjecture - Mathematics > Rings and Algebras

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Abstract: Let $\K$ denote a field and let $V$ denote a vector space over $\K$ withfinite 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 $\lbraceV i brace {i=0}^d$ of the eigenspaces of $A$ such that $A^* V i \subseteqV {i-1} + V {i} + V {i+1}$ for $0 \leq i \leq d$, where $V {-1}=0$ and$V {d+1}=0$; iii there exists an ordering $\lbrace V^* i brace {i=0}^\delta$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 $AW\subseteq W$, $A^* W \subseteq W$, $W eq 0$, $W eq V$. We call such a paira {\it tridiagonal pair} on $V$. It is known that $d=\delta$ and for $0 \leq i\leq d$ the dimensions of $V i$, $V {d-i}$, $V^* i$, $V^* {d-i}$ coincide. Wesay the pair $A,A^*$ is {\it sharp} whenever $\dim V 0=1$. It is known that if$\K$ is algebraically closed then $A,A^*$ is sharp. A conjecturedclassification of the sharp tridiagonal pairs was recently introduced by T. Itoand the second author. Shortly afterwards we introduced a conjecture, calledthe {\em $\mu$-conjecture}, which implies the classification conjecture. Inthis paper we show that the $\mu$-conjecture holds in a special case called$q$-Racah.

Author: Kazumasa Nomura, Paul Terwilliger

Source: https://arxiv.org/