Leonard pairs having zero-diagonal TD-TD form

2015/03/18 by Kazumasa Nomura, Nomura, Kazumasa
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA

paper · pdf · doi:10.48550/arxiv.1503.05262

arxiv created 2015/03/18 · arxiv updated 2015/03/19

Abstract

Fix an algebraically closed field \mathbbF and an integer n ≥ 1. Let Mat_n(\mathbbF) denote the \mathbbF-algebra consisting of the n × n matrices that have all entries in \mathbbF. We consider a pair of diagonalizable matrices in Mat_n(\mathbbF), each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in Mat_n(\mathbbF). In the present paper, we find all Leonard pairs A,A^* in Mat_n(\mathbbF) such that each of A and A^* is irreducible tridiagonal with all diagonal entries 0. This solves a problem given by Paul Terwilliger.

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