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 a 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.

arXiv comment32 pages; sequel to arXiv:1704.02668

References in corpus (8)

• Stein's method and the rank distribution of random matrices over finite fields
• The average size of the kernel of a matrix and orbits of linear groups
• Skolem-Noether for nilpotent products
• Uniform cell decomposition with applications to Chevalley groups
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H
• General Bilinear Forms
• Division, adjoints, and dualities of bilinear maps
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties

Cited by in corpus (6)

• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties
• Univariate and bivariate zeta functions of unipotent group schemes of type $G$
• Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups
• A spectral theory for transverse tensor operators
• Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves
• Smooth cuboids in group theory