An Introduction to Group Representations and Orthogonal Polynomials

An elementary non-technical introduction to group representations and orthogonal polynomials is given. Orthogonality relations for the spherical functions for the rotation groups in Euclidean space (ultraspherical polynomials), and the matrix elements of SU(2) (Jacobi polynomials) are discussed. A general theory for finite groups acting on graphs, giving a finite set of discrete orthogonal polynomials is given. Explicit examples include graphs giving the Krawtchouk and Hahn polynomials. Introduction. The purpose of this paper is to present a friendly, non-technical introduction to group representations and orthogonal polynomials. No previous knowledge of group representations is assumed, but a familiarity with orthogonal polynomials is assumed. In particular, this paper emphasizes the classical orthogonal polynomials and their relationship to groups. Other classical special functions can also be studied in this way, e.g. Vilenkin [19] or Miller [14] (which is more elementary). §I of this paper could be considered as a short introduction to the sections of those books relevant to orthogonal polynomials. More modern work on continuous groups and the related analysis has been done by Koornwinder [11] and Dunkl [9]. Some very recent work concerns orthogonal polynomials in several variables [13]. It was not realized until the early 1970’s that finite groups could be related to classical orthogonal polynomials. The pioneering work was done by Dunkl [8],[9] and Delsarte [5],[6],[7]. §II is an introduction to the general theory of finite groups. This theory can be generalized to association schemes, which consider relations on a finite set with certain properties. An extensive theory of association schemes can be found in [3] and [5]. A very elementary introduction is given in [16]. A survey of recent work and important problems is given in [4], and in Bannai’s paper [2] in this volume. Notation. The classical orthogonal polynomials can be expressed as hypergeometric series. We will use the usual notation for these series (see [15]). We will be most concerned about three sets of polynomials: Jacobi, Krawtchouk and Hahn. The Jacobi polynomials (Ja) P (α,β) n (x) = (α+ 1)n n! 2F1 ( −n, n+ α+ β + 1; 1− x α+ 1 2 )