Tridiagonal pairs of $q$-Racah type

Abstract

Let $K$ denote an algebraically closed 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 {\it tridiagonal pair} on $V$. It is known that $d=\delta$. For $0\le i\le d$ let ${\theta }_i$ (resp. ${\theta }_i^{\ast }$) denote the eigenvalue of $A$ (resp. $A^{\ast }$) associated with $V_i$ (resp. $V_i^{\ast }$). The pair $A,A^{\ast }$ is said to have {\it $q$-Racah type} whenever ${\theta }_i=a+b{q}^{2i-d}+c{q}^{d-2i}$ and ${\theta }_i^{\ast }=a^{\ast }+b^{\ast }{q}^{2i-d}+c^{\ast }{q}^{d-2i}$ for $0\le i\le d$, where $q,a,b,c,a^{\ast },b^{\ast },c^{\ast }$ are scalars in $K$ with $q,b,c,b^{\ast },c^{\ast }$ nonzero and $q^2\notin \{1,-1\}$. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over $K$ that have $q$-Racah type. Our proof involves the representation theory of the quantum affine algebra $U_q({\widehat{\mathfrak{sl}}}_2)$.

30 pages

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Cited by in corpus (3)

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