Group representations and multinomial combinatorics of the icosahedral symmetry

The icosahedral symmetry is one of the most intriguing symmetries, as it not only presents a challenge but it appears in many fullerenes and high energetic materials such as the dodecahedral N-20. We have considered the combinatorics of all irreducible representations of the icosahedral symmetry for a number of multinomial partitions for vertex, face and edge colourings in this work. We have constructed the combinatorial tables for all irreducible representations for various multinomial partitions of colourings for the vertices, edge and faces of the icosahedron. These techniques should have important applications to enumerations and spectroscopy of fullerenes and high-energy materials such as N-20.