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

TL;DR

This paper introduces duality functors between module representation categories, generalizing previous duality phenomena and analyzing their effects on kernel sizes, with applications to class numbers and zeta functions of p-groups.

Contribution

It defines new duality functors indexed by the symmetric group and demonstrates their tame effects on average kernel sizes, extending prior duality results.

Findings

Duality functors have tame effects on average kernel sizes.

Framework generalizes duality phenomena in module representations.

Applications to class numbers and conjugacy class zeta functions.

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.