Groups, graphs, and hypergraphs: average sizes of kernels of generic matrices with support constraints

Abstract

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 F q -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.

On Denert's statistic

“…By results going back to MacMahon, our equidistribution result also implies a remarkable identity involving Hadamard products of genus zeta functions of local hereditary orders. Similar identities, also involving Eulerian or Euler-Mahonian polynomials, have appeared in recent work on so-called ask zeta functions [18,19] and zeta functions associated with quiver representations [15].…”

Enumerating conjugacy classes of graphical groups over finite fields

“…We call f Γ (X) the class-counting polynomial of Γ. In [23], Theorem 1.2 is derived from a more general uniformity result [23,Cor. B] for class-counting zeta functions associated with graphical group schemes; see §8.1.…”

“…Note that every ring map R → R induces an evident group homomorphism G Γ (R) → G Γ (R ). We will see in §2.3 that the resulting group functor G Γ represents the graphical group scheme associated with Γ as defined in [23]. The isomorphism type of G Γ does not depend on the chosen ordering of the vertices of Γ.…”

“…For any graph Γ, the size of each conjugacy class of G Γ (F q ) is of the form q i ; see Proposition 2.3(i) As indicated in [23, §8.5], the methods underpinning Theorem 1.2 can be used to strengthen said theorem: each cc q i (G Γ (F q )) is a polynomial in q with rational coefficients. While constructive, the proof of Theorem 1.2 in [23] relies on an elaborate recursion. In particular, no explicit general formulae for the numbers cc q i (G Γ (F q )) or the polynomial f Γ (X) have been previously recorded.…”

“…In §2, we relate the definition of graphical groups from §1.2 to that from [23]. Introduced in [23], adjacency modules are modules over polynomial rings whose specialisations are closely related to conjugacy classes of graphical groups. In §3, we determine the dimensions of such specialisations over fields.…”