football representation theory
Unless
you never touched a football in your life (that’s a soccer-ball
for those of you with an edu account) you will know that the world
championship in Germany starts tonight. In the wake of it, the field of
‘football-science’ is booming. The BBC runs its The
Science of Football-site and did you know the following?
Research indicates that watching such a phenomenon is not only exciting, it can be good for our health too. The Scottish researchers found that there were 14% fewer psychiatric admissions in the weeks after one World Cup than before it started.
But, would you believe that some of the best people in the field (Kostant and Sternberg to name a few) have written papers on the representation theory of a football? Perhaps this becomes more plausible when you realize that a football has the same shape as the buckyball aka Carbon60. Because the football (or buckyball) is a truncated icosahedron, its symmetry group is $A5$, the smallest of all simple groups and its representations explain some physical properties of the buckyball. Some of these papers are freely available and are an excellent read. In fact, I’m thinking of using them in my course on representations of finite groups, nxt year. Mathematics and the Buckyball by Fan Chung and Schlomo Sternberg is a marvelous introduction to representation theory. Among other things they explain how Schur’s lemma, Frobenius reciprocity and Maschke’s theorem are used to count the number of lines in the infra red buckyball spectrum! The Graph of the Truncated Icosahedron and the Last Letter of Galois by Bertram Kostant explains the observation, first made by Galois in his last letter to Chevalier, that $A{5} = PSL2(\mathbb{F}5)$ embeds into $PSL{2}(\mathbb{F}{11})$ and applies this to the buckyball.
In effect, the model we are proposing for C60is such that each carbon atom can be labeled by an element of order 11 in PSl(2,11) in such a fashion that the carbon bonds can be expressed in terms of the group structure of PSl(2,11). It will be seen that the twelve pentagons are exactly the intersections of M with the twelve Borel sub- groups of PSl(2,11). (A Borel subgroup is any subgroup which is conjugate to the group PSl(2,11) defined in (2).) In particular the pentagons are the maximal sets of commuting elements in M. The most subtle point is the natural existence of the hexagonal bonds. This will arise from a group theoretic linkage of any element of order 11 in one Borel subgroup with a uniquely defined element of order 11 in another Borel subgroup.These authors consequently joined forces to write Groups and the Buckyball in which they give further applications of the Galois embeddings to the electronic spectrum of the buckyball. Another account can be found in the Master Thesis by Joris Mooij called The vibrational spectrum of Buckminsterfullerene - An application of symmetry reduction and computer algebra. Plenty to read should tonight’s match Germany-Costa Rica turn out to be boring…