Vibrational modes of Platonic molecules

Imagine a molecule whose atoms sit at the vertices of a Platonic solid when in equilibrium. Model the interactions of the atoms by springs along the edges of the solid. Displace the atoms slightly from the equilibrium position and the molecule will vibrate. The aim of the project is to understand the vibrational modes of such a molecule in terms of the representation theory of the group of symmetries of the platonic solid.

This requires learning about characters of finite groups, for which the first three references below are a typical source: first two chapters in Fulton and Harris, chapter 3 in Miller or chapters 2 and 5 in Serre.

It is conceivable that this project might involve some computer work.

Extensions

This project could be extended to a Combined Degree (Mathematics and Physics) Project , by adding more complicated examples, like the bucky ball, covered in some of the papers listed below.

References

1. Representation theory: a first course
   William Fulton and Joe Harris
2. Symmetry groups and their applications
   Willard Miller
3. Linear representations of finite groups
   Jean-Pierre Serre
4. The representation theory of Buckminsterfullerene
   G James
   J. Algebra 167 (1994) 803-820.
5. The graph of the truncated icosahedron and the last letter of Galois
   B Kostant
   Notices Am. Math. Soc. 42 (1995) 959-968.
6. Mathematics and the Buckyball
   F Chung and S Sternberg
   Amer. Scientist 81 (1993) 56-71.