Edmund Heng (IHÉS)

Abstract: Many properties of Coxeter groups can be deduced via what is known as the Tits representation, which (faithfully) realises each Coxeter group as a linear reflection group over the real numbers. In constructing the Tits representation, an important role is played by the numbers 2cos(π/m), which appear in the associated standard bilinear form (or the Cartan matrix) of the Coxeter group. As it turns out, these numbers 2cos(π/m) also play an important role in a rather far-distant land of quantum mathematics: they appear in the theory of (Frobenius-Perron) dimension of fusion rings (or fusion categories), which serves as a generalisation of dimensions for vector spaces.

In this talk, I will build on this rather naive looking numerical observation and introduce a slightly generalised version of the Tits representation, which essentially replaces the ring of real numbers with a fusion ring. I will show how these faithful representations naturally provide embeddings of hyperplane complements associated to Coxeter groups, and discuss some its relation to the K(π,1) conjecture for Artin-Tits groups. If time allows, I will also provide a big picture of a personal long-term goal that aims to extend (and complement) Coxeter theory in (and using) quiver theory, Kac-Moody algebras etc.