Groups, Graphs, and Hypergraphs: Average Sizes of Kernels of Generic Matrices with Support Constraints (Memoirs of the American Mathematical Society)

Rossmann, Tobias/ Voll, Christopher

American Mathematical Society（2024/05発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 120 p.
• 言語 ENG
• 商品コード 9781470468682
• DDC分類 512.2

Full Description

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of Fq-points of the groups under consideration depend polynomially on q. Our approach combines group theory, graph theory, toric geometry, and p-adic integration.

Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.

Contents

1. Introduction
2. Ask zeta functions and modules over polynomial rings
3. Modules and module representations from (hyper)graphs
4. Modules over toric rings and associated zeta functions
5. Ask zeta functions of hypergraphs
6. Uniformity for ask zeta functions of graphs
7. Graph operations and ask zeta functions of cographs
8. Cographs, hypergraphs, and cographical groups
9. Further examples
10. Open problems