Balanced Leonard pairs
type: Item
instance of: scholarly article
title: Balanced Leonard pairs
zbMATH Open document ID: 1119.05110
published in: Linear Algebra and its Applications
publication date: 2007-02-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0506219
review text: Let \(V\) denote a finite-dimensional vector space over any field, dim\(\,V= d+1>0\). A Leonard pair \((A, A^*)\) is a pair of linear transformations \(A: V\to V\) and \(A^*: V\to V\) with the following properties: (i) There exists a basis \((v_i^*)\) of \(V\) such that the matrices of \(A\) and \(A^*\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. (ii) There exists a basis \((v_i)\) of \(V\) such that the matrices of \(A^*\) and \(A\) with respect to this basis are irreducible tridiagonal and diagonal, respectively. Denote by \(a_i\) the \(i\)th entry in the main diagonal of the tridiagonal matrix representing \(A\) with respect to \((v_i^*)\), where \(0\leq i\leq d\). The scalars \(a_i^*\) are defined accordingly. The Leonard pair \((A, A^*)\) is said to be balanced if \(a_i=a_{d-i}\) and \(a_i^*=a_{d-i}^*\) for \(0\leq i\leq d\). The Leonard pair \((A, A^*)\) is said to be essentially bipartite (essentially dual bipartite) if \(a_i\) (\(a_i^*\)) is independent of \(i\) for \(0\leq i\leq d\). The authors show that \(a_0=a_d\) if, and only if, \(a_0^*=a_d^*\). Furthermore, the following assertions are shown to be equivalent: (i) \(a_0=a_d\) and \(a_1=a_{d-1}\); (ii) \(a_0^*=a_d^*\) and \(a_1^*=a_{d-1}^*\); and (iii) \((A, A^*)\) is balanced. For \(d\not=2\), the authors also prove that if \((A,A^*)\) is balanced then the pair \((A, A^*)\) is essentially bipartite or dual bipartite.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D45
zbMATH DE Number: 5126050
zbMATH Keywords: Leonard pair
zbMATH Keywords: Terwilliger algebra
zbMATH Keywords: Askey scheme
zbMATH Keywords: q-Racah polynomial
author: Kazumasa Nomura
author: Paul Terwilliger
MaRDI profile type: Publication
OpenAlex ID: W1990571726
arXiv ID: math/0506219
cites work: Tight distance-regular graphs and the \(Q\)-polynomial property
cites work: Two linear transformations each tridiagonal with respect to an eigenbasis of the other
cites work: Leonard pairs from 24 points of view.
cites work: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
DOI: 10.1016/J.LAA.2006.06.025
description: scientific article