Determining the representations of a finite group over the real numbers is an old problem, and is elegantly solved using that complex representations fall into one of three cases – corresponding to the three finite division algebras over R. A natural generalisation, phrased in terms of complex anti-linear maps, was first considered by the physicist Eugene Wigner due to its applications in the physical context of time reversal in quantum mechanics. Astoundingly, the trichotomy that exists between R and C completely generalises to this setting, and results in what Freeman Dyson coined “The Tenfold Way” when the classical theory and its generalisation are considered simultaneously. Another completely surprising application is a method of counting the number of square roots of group elements.

In the talk we will explore the phenomena of the classical theory and its generalisation, along with what this all means for the A_n and S_n.