Modular Leonard triples
type: Item
instance of: scholarly article
title: Modular Leonard triples
zbMATH Open document ID: 1124.15001
published in: Linear Algebra and its Applications
publication date: 2007-06-26T00:00:00Z
review text: Let \(\mathbb{K}\) denote a field. A square matrix over \(\mathbb{K}\) is said to be tridiagonal whenever every nonzero entry appears on the diagonal, the superdiagonal, or the subdiagonal. A tridiagonal matrix is irreducible whenever the entries on the sub- and superdiagonals are all nonzero. Let \(V\) denote a vector space over \(\mathbb{K}\) of finite positive dimension. An ordered triple \(A\), \(A^*\), \(A^\diamondsuit\) of linear operators on \(V\) is said to be a Leonard triple whenever for each \(B\in\{A, A^*, A^\diamondsuit\}\), there exists a basis of \(V\) with respect to which the matrix representing \(B\) is diagonal and the matrices representing the other two operators are irreducible tridiagonal. A Leonard triple \(A\), \(A^*\), \(A^\diamondsuit\) is said to be modular whenever for each \(B\in\{A,A^*,A^\diamondsuit\}\), there exists an antiautomorphism of \(\text{End}(V)\) which fixes \(B\) and swaps the other two operators. The paper classifies the modular Leonard triples up to isomorphism.
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5167493
zbMATH Keywords: antiautomorphisms
zbMATH Keywords: tridiagonal
zbMATH Keywords: irreducible
zbMATH Keywords: Leonard triple
author: Brian Curtin
reviewed by: Rodica Covaci
MaRDI profile type: Publication
full work available at URL: j.laa.2007.02.024
OpenAlex ID: W2025201122
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DOI: 10.1016/J.LAA.2007.02.024
description: scientific article; zbMATH DE number 5167493