Algebra Seminar Talk by Christopher Voll (Bielefeld University)

This Wednesday 22nd November 2023, we had the pleasure to digitally welcome Christopher Voll (Bielefeld University) to the Charlotte Scott Centre for Algebra, and to hear about his work on Groups, graphs, hypergraphs — average sizes of kernels of generic matrices and conjugacy classes of finite p-groups.

Abstract: A fundamental invariant of a finite group is the number of its
conjugacy classes or, equivalently, its irreducible complex
characters. The determination of this “class number” is, in general, a
difficult problem. The variation of class numbers in natural families
is the object of central conjectures in the theory of finite p-groups,
such as G. Higman’s famous conjecture on the polynomiality of class
numbers of full unitriangular matrix groups over finite fields.

I report on joint work with Tobias Rossmann. We develop a theory of
average sizes of kernels of generic matrices with support constraints
encoded by graphs and hypergraphs. As an application of this theory,
we prove strong uniformity theorems about the variation of the class
numbers of finite “graphical groups”, certain finite p-groups
associated with graphs in a canonical way. These uniformity
(i.e. polynomiality) theorems are in striking contrast to
algebro-geometric results of Belkale and Brosnan about the rank
distribution of symmetric matrices with support conditions over finite
fields: the exact rank distribution is — in a precise, technical
sense — arbitrarily wild.

Methodologically, our project combines ideas from graph and group
theory, but also from toric geometry and p-adic integration theory,
and has strong algorithmic aspects. (https://arxiv.org/abs/1908.09589)