Orthogonal polynomials of compact simple Lie groups. (English) Zbl 1250.33016
Special functions of mathematical physics are in fact matrix elements of representations of Lie groups, and recent multivariate generalizations of classical hypergeometric orthogonal polynomials are based on root systems of simple Lie groups/algebras. The authors propose a recursive algebraic construction of two infinite families of orthogonal multivariate polynomials related to orbit functions of simple Lie groups of rank \(n\). Their method is based on the decomposition of products of Weyl group orbits and on the basic properties of the characters of irreducible finite dimensional representations. The paper focuses on polynomials obtained by substitution of variables, mimicking Weyl’s method for the construction of finite-dimensional representations from \(n\) fundamental representations. A number of multivariate generalizations of classical Chebyshev polynomials are available in the literature. The authors show in all details how Chebyshev polynomials appear as particular case of multivariate polynomials. Recurrence relations are shown for the Lie groups of types \(A_1, A_2, A_3, C_2, C_3, G_2\), and \(B_3\) together with lowest polynomials.
Reviewer: Nicolae Cotfas (Bucureşti)
MSC:
| 33C52 | Orthogonal polynomials and functions associated with root systems |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 33D52 | Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) |
| 22E46 | Semisimple Lie groups and their representations |
| 33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |
| 05E05 | Symmetric functions and generalizations |
Keywords:
orthogonal polynomials; simple Lie groups; orthogonal multivariate polynomials; Weyl group; Chebyshev polynomialsReferences:
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