Krawtchouk polynomials, the Lie algebra \(\mathfrak{sl}_2\), and Leonard pairs
type: Item
instance of: scholarly article
title: Krawtchouk polynomials, the Lie algebra \(\mathfrak{sl}_2\), and Leonard pairs
zbMATH Open document ID: 1261.33001
published in: Linear Algebra and its Applications
publication date: 2012-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.1645
reviewed by: Kevin Coulembier
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 6034535
zbMATH Keywords: Krawtchouk polynomials
zbMATH Keywords: Leonard pair
author: Kazumasa Nomura
author: Paul Terwilliger
MaRDI profile type: Publication
OpenAlex ID: W2033141712
arXiv ID: 1201.1645
Wikidata QID: Q115344695
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DOI: 10.1016/J.LAA.2012.02.006
review text: In this paper the authors give a self-contained exposition on the appearance of Krawtchouk polynomials and Leonard pairs in the study of finite-dimensional representations of the Lie algebra \(\mathfrak{sl}(2)\). Many of the results were previously known but they are presented here as a tutorial.NEWLINENEWLINEDenote by \(h\) the standard Cartan element of \(\mathfrak{sl}(2)\), \(e\) the positive and \(f\) the negative root vector. It is well known that the matrices corresponding to the action of \(h\) and \(e+f\) are conjugated by the action of a matrix with entries given in terms of the Krawtchouk polynomials \(K_i(-;1/2,N)\). The first result in the paper is that this property can be extended to any pair of normalized semisimple elements of \(\mathfrak{sl}(2)\) (elements in the \(\mathrm{SL}(2)\)-orbit of \(h\)). There, the corresponding matrix elements are expressed in terms of the Krawtchouk polynomials \(K_i(-;p,N)\) with \(p\) determined by the action of the Killing form on the pair.NEWLINENEWLINEThen the notion of Leonard pair is introduced and explored, in particular the Leonard pairs of Krawtchouk type. It follows that the matrices corresponding to the action of a pair of normalized semisimple elements of \(\mathfrak{sl}(2)\) on an irreducible finite-dimensional representation form a Leonard pair of Krawtchouk type. Reversely, any pair of matrices that is a Leonard pair of Krawtchouk type generates an irreducible representation of \(\mathfrak{sl}(2)\).
description: scientific article; zbMATH DE number 6034535