Groups, Graphs, and Hypergraphs: Average Sizes of Kernels of Generic Matrices with Support Constraints

Tobias Rossmann, Christopher Voll (Autoren)

Buch | Softcover

120 Seiten

2024
American Mathematical Society (Verlag)
978-1-4704-6868-2 (ISBN)

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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.

Tobias Rossmann, University of Galway, Ireland. Christopher Voll, Universitat Bielefeld, Germany.

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

Erscheinungsdatum	05.04.2024
Reihe/Serie	Memoirs of the American Mathematical Society ; Volume: 294 Number: 1465
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	118 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Graphentheorie
ISBN-10	1-4704-6868-9 / 1470468689
ISBN-13	978-1-4704-6868-2 / 9781470468682
Zustand	Neuware