Generating functions for the Coxeter group

Abstract

Generating functions are found for decomposition of the space of homogeneous polynomials of any degree in four variables into the direct sum of subspaces irreducible under the group , the non-crystallographic Coxeter group of order 14 400. The four variables are coordinates of a vector from the defining four-dimensional representation space of the group. As the defining representation we consider any of the four non-equivalent irreducible representations of of dimension four. Analogous generating functions for the binary icosahedral group of order 120 (generated by reflections, i.e. a subgroup of but not of ) and for the dihedral group of order 10 are also rederived and shown. The groups are naturally related by .