The average size of the kernel of a matrix and orbits of linear groups, II: duality

Abstract

Define a module representation to be a linear parameterisation of a collection of module homomorphisms over a ring. Generalising work of Knuth, we define duality functors indexed by the elements of the symmetric group of degree three between categories of module representations. We show that these functors have tame effects on average sizes of kernels. This provides a general framework for and generalisation of duality phenomena previously observed in work of O'Brien and Voll and in the predecessor of the present article. We discuss applications to class numbers and conjugacy class zeta functions of p-groups and unipotent group schemes, respectively.

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

“…certain combinatorially defined subgroups of U n (F q ). This connection also occurs in previous work [52,57,60] of both authors. Moreover, it turns out that if we are willing to exclude small exceptional characteristics, the study of k(G(F q )) for group schemes G U n as above essentially reduces to those of class 2.…”

Hessian matrices, automorphisms of p-groups, and torsion points of elliptic curves

“…Arithmetic properties of finite p-groups arising as groups of F-rational points of group schemes defined in terms of matrices of linear forms are also a common theme of [2,20,22,24] (with a view towards the enumeration of conjugacy classes) and [3] (with a view towards faithful dimensions; cf. also Sect.…”

“…7.6 (ii)] of A(X). The fact that the bottom part of A(X) is, up to a sign, equal to its top part (and not its •-dual in the sense of [22, § 4.1]) reflects the symmetry of B; see [22,Prop. 4.12].…”

Enumerating conjugacy classes of graphical groups over finite fields

“…Beginning with work of du Sautoy [5], these and closely related series enumerating conjugacy classes have recently been studied, see [1,14,15,[21][22][23] For each graph Γ, there exists a rational function WΓ (X, Y ) ∈ Q(X, Y ) with the following property: for each compact discrete valuation ring O with residue field size q, we have ζ k G Γ ⊗O (s) = WΓ (q, q −s ).…”