A classification of sharp tridiagonal pairs

Abstract

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 $A^{\ast }:V\to V$ 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 $d=\delta$ and for $0\le i\le d$ the dimensions of $V_i,V_{d-i},V_i^{\ast },V_{d-i}^{\ast }$ coincide. The pair $A,A^{\ast }$ is called {\it sharp} whenever $\dim V_0=1$. It is known that if $F$ is algebraically closed then $A,A^{\ast }$ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the $\mu$-conjecture.

36 pages

References in corpus (12)

• Orthogonal Polynomials from Hermitian Matrices
• The q-deformed analogue of the Onsager algebra: Beyond the Bethe ansatz approach
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
• Some algebra related to $P$-and $Q$-polynomial association schemes
• The Drinfel'd polynomial of a tridiagonal pair
• Quasi-Linear Algebras and Integrability (the Heisenberg Picture)
• Finite-dimensional irreducible modules for the three-point ${\mathfrak{sl}}_2$ loop algebra
• How to sharpen a tridiagonal pair
• Towards a classification of the tridiagonal pairs
• Tridiagonal pairs of shape (1,2,1)
• Mock Tridiagonal Systems

Cited by in corpus (5)

• The $q$-Onsager Algebra and the Universal Askey-Wilson Algebra
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• Evaluation modules for the $q$-tetrahedron algebra
• The algebra $U_q({\mathfrak{sl}}_2)$ in disguise
• Two commuting operators associated with a tridiagonal pair