Pure Colloquium Seminar, Representative growth of simple Lie groups, Alex Stasinski (Durham) Seminar
- Time:
- 15:00 - 16:00
- Date:
- 12 January 2018
- Venue:
- Room 10031, Lecture Theatre 10C, Building 54, University of Southampton, Highfield Campus, SO17 1BJ
For more information regarding this seminar, please email Dr Ashot Minasyan at A.Minasyan@southampton.ac.uk .
Event details
For a given group G, representation growth is the study of the asymptotic properties of the sequence r_i(G) of the number of irreducible (complex) representations of G of dimension i. One of the main tools in this study is the representation zeta function of G, which is the Dirichlet series associated to the sequence r_i(G). When G is a simple compact Lie group, this is known as the Witten zeta function.
In a seminal paper, M. Larsen and A. Lubotzky initiated a systematic study of representation zeta functions and emphasised the abscissa of convergence as the primary object of study. This is the real number which defines the half-plane of convergence of the zeta function and also gives the rate of polynomial growth of r_i(G). Larsen and Lubotzky proved that the abscissa of a Witten zeta function is r/k, where r is the rank of G and k is the number of positive roots. I will present some of the ideas in a new proof of this result, and explain how it leads to a generalisation beyond Witten zeta functions and root systems. In particular, this goes some way towards explaining where the constant r/k comes from.
This is joint work with J. Häsä.
Speaker information
Alex Stasinski, Durham.