Krawtchouk polynomials, the Lie algebra \(\mathfrak{sl}_2\), and Leonard pairs (Q417601)

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 \(\mathfrak{sl}(2)\).