Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

TL;DR

This paper explores the deep connections between Krawtchouk polynomials, the Lie algebra rak{sl}_2, and Leonard pairs, providing elementary proofs and a comprehensive tutorial suitable for graduate students and researchers.

Contribution

It offers a unified, elementary account of how Krawtchouk polynomials relate to rak{sl}_2} modules and Leonard pairs with Krawtchouk type, including new proofs of classical properties.

Findings

Elementary proofs of Krawtchouk polynomial properties

Clarification of the relationship between Leonard pairs and rak{sl}_2} modules

Identification of Krawtchouk type Leonard pairs

Abstract

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary but comprehensive account of how the following are related: (i) Krawtchouk polynomials; (ii) finite-dimensional irreducible modules for the Lie algebra ${\mathfrak{sl}_2}$; (iii) a class of Leonard pairs said to have Krawtchouk type. Along the way we obtain elementary proofs of some well-known facts about Krawtchouk polynomials, such as the three-term recurrence, the orthogonality, the difference equation, and the generating function. The paper is a tutorial meant for a graduate student or a researcher unfamiliar with the above topics.