GENERATING-FUNCTIONS FOR THE COXETER GROUP H-4

Generating functions are found for decomposition of the space of homog eneous polynomials of any degree in four variables into the direct sum of subspaces irreducible under the group H-4, the non-crystallographi c Coxeter group of order 14400. The four variables are coordinates of a vector from the defining four-dimensional representation space of th e group. As the defining representation we consider any of the four no n-equivalent irreducible representations of H-4 of dimension four. Ana logous generating functions for the binary icosahedral group H-3 of or der 120 (generated by reflections, i.e. a subgroup of O(3) but not of SO(3)) and for the dihedral group H-2 of order 10 are also rederived a nd shown. The groups are naturally related by H-4 superset of H-3 supe rset of H-2.