Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

Abstract

Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤ 3 are explicitly studied. We derive the polynomials of simple Lie groups B 3 and C 3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

On immanant functions related to Weyl groups of A n

“…We obtain the cosine and sine functions of three complex variables, see [22], and three new functions. The implicit forms in the terms of Weyl groups are given below:…”

“…22 a 33 a 44 , C 2 = a 11 a 22 a 34 a 43 + a 11 a 23 a 32 a 44 + a 11 a 24 a 33 a 42 + a 12 a 21 a 33 a 44 + a 13 a 22 a 31 a 44 + a 14 a 22 a 33 a 41 , C 3 = a 12 a 21 a 34 a 43 + a 13 a 24 a 31 a 42 + a 14 a 23 a 32 a 41 , C 4 = a 11 a 23 a 34 a 42 + a 11 a 24 a 32 a 43 + a 13 a 22 a 34 a 41 + a 14 a 22 a 31 a 43 + a 12 a 24 a 33 a 41 + a 14 a 21 a 33 a 42 + a 12 a 23 a 31 a 44 + a 13 a 21 a 32 a 44 , C…”

Orthogonal Polynomials of Compact Simple Lie Groups

“…In general, for a given group G, most of the properties of orbit functions come from the Weyl group and the irreducible character of G or from the basic notions of the representation theory. This fact makes multivariate polynomials obtained from orbit functions attractive and handy for a wide range of applications, such as discrete and interpolation problems, cubature formulas [35], reduction of polynomials [40], etc.…”

“…All, but two last cases, contain both generic and additional recursions. For groups C 3 and B 3 we obtain only generic recurrence relations and lowest polynomials necessary to solve them are available in [40].…”

Multidimensional Hybrid Boundary Value Problem

“…The solutions are presented as expansions into a series of special functions that satisfy required conditions at the (n − 1)-dimensional boundaries of F . The recent discovery of special functions [5,10,11,16,19,23] makes realization of this idea easy and straightforward in any dimension. The new functions, called 'multidimensional hybrids', satisfy the Dirichlet boundary condition on some parts of the boundary F and Neumann on the remaining ones.…”