Symmetries of Regular Geometrical Objects

Abstract

When we think about symmetries, regular geometrical objects are the ones which come first to mind. While it is true that the concept of symmetry has a much wider meaning in mathematics and physics, the importance of symmetries of basic geometrical objects should not be taken lightly. They play pivotal role in the study of many kinds of systems, from molecules to crystals and beyond. In this chapter, we discuss the symmetry groups associated with some regular geometrical object in two and three dimensions, and the representations of those groups. SYMMETRIES OF REGULAR POLYGONS We have mentioned before that the symmetry group of a regular polygon with n vertices is called D n , and that it contains 2 n elements, among which there is of course the identity element, n − 1 non-trivial rotations and n reflections. In Section 8.9.5, we discussed the conjugacy class structure of the D n groups. Taking the cue from there, we discuss here the irreducible representations of D n groups. As is clear from the discussion of Section 8.9.5, this discussion will have to be different for odd and even values of n . Representations of Dn for odd n We take n = 2 r +1, where r is an integer. THEOREM 10.1 The group Dn, for n = 2 r + 1 , admits two 1-dimensional and r 2-dimensional irreps. PROOF: We saw in Section 8.9.5 that the number of conjugacy classes is r + 2. The total number of elements is 2 n = 4 r +2. There is only one way that the equation for the dimensionalities of irreps, Eq. (9.86) can be satisfied, and the solution is given in the statement of this result. The characters for the 1-dimensional irreps are obvious. One of these is the trivial representation for which all characters are equal to 1. The other has χ = +1 for the even permutations and χ = −1 for the odd permutations. Before trying to give the characters of the 2-dimensional irreps for general D n , let us try to find the character table for D 5 as an example.