Two linear transformations each tridiagonal with respect to an eigenbasis of the other

2004/06/27 by Paul Terwilliger · 2 citations
Mathematics · Physics and Astronomy · #math.RA #math-ph #math.MP #msc:17B37

paper · pdf

published as Linear Algebra Appl. 330 (2001), 149--203

arxiv created 2004/06/27 · arxiv updated 2009/12/01

Abstract

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A:V→ V and A^*:V→ V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal, and the matrix representing A^* is irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A^* is diagonal, and the matrix representing A is irreducible tridiagonal. We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. We discuss how Leonard systems correspond to the q-Racah and related polynomials from the Askey scheme.

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• Spin Leonard pairs and the zero diagonal space
• Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way

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