Linear Representations of Finite Groups and The Ideal Theoretical Construction of G-Invariant Cubature Formulas

Abstract

Under some conditions the common zeros of some sufficiently orthogonal polynomials may be chosen as the nodes of a cubature formula (Moller). Once the nodes are known the weights are determined by the solution of a system of linear equations. It is shown how in this approach the theory of linear representations of finite groups exploites the symmetry which is an additionally required property of cubature formulas for regions such as the triangle, square or the tetrahedron. By means of this theory reasonable parameter dependent orthogonal polynomials are chosen and equations for these parameters are determined from some necessary conditions, the existence of the so-called syzygies.