By Tammo tom Dieck

This publication is a jewel– it explains very important, invaluable and deep themes in Algebraic Topology that you simply won`t locate in different places, conscientiously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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The 80-vertex structure is formed by combination of the O20 orbit and an O60 orbit of Ih point symmetry to realize the three-valent fullerene. The permutation character over the full set of 80 vertices is the direct sum σ = 2Ag + T1g + 3T1u + T2g + 3T2u + 3Gg + 3Gu + 4Hg + 2Hu because of extra components due to the O20 orbit permutation character. 1 would be required, but it is simpler to carry out a two-orbit analysis, taking two copies of 1T1u , the first over 60 vertices and the second over 20, with the third T1u set being the projections of 2T1u on the vertices of O60 .

Molecular frameworks with the shapes of each Platonic solid are known. In symmetry terms, the Platonic solids split into two families: the tetrahedron, the cube and the octahedron, which have cubic symmetry, and the icosahedron and the dodecahedron, which have icosahedral symmetry. A further set of semi-regular polyhedra, many of which are also important in Chemistry, follows on relaxation of the requirement for equivalent polygonal faces. The thirteen Archimedean polyhedra have equivalent vertex figures on all vertices and all faces remain planar and equilateral, but are of two or three distinct kinds.

26 The central polynomial functions of 5Hu symmetry, which are orthonormal on the unit sphere and, which along with 1Hu , 2Hu , 3Hu and 5Hu are required to make basis functions for all of the 25 functions of Hu symmetry in the regular representation of Ih . e. m, n, p = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, . . 26, of the 5th copy of five polynomial functions, which, on the unit sphere, are mutually orthonormal to one another and to the 20 other polynomial functions of this irreducible symmetry forming the 1st , 2nd , 3rd and 4th sets of functions also of this symmetry, displayed when the other buttons on the keypad labelled with this symmetry are selected by a mouse click.

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Algebraic Topology and Tranformation Groups by Tammo tom Dieck

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