Genotypes of irreducible representations of finite p-groups

TL;DR

This paper introduces the concept of genotype for irreducible representations of finite p-groups, linking genetic invariants to the genotype and applying this to count Galois conjugacy classes.

Contribution

It defines the genotype of irreducible representations for finite p-groups and explores its invariants, extending previous special cases and providing applications to Galois conjugacy class counting.

Findings

Genetic invariants are exactly the invariants of the genotype.

The genotype is a Roquette p-group associated with an irrep.

Application to counting Galois conjugacy classes.

Abstract

For any characteristic zero coefficient field, an irreducible representation of a finite $p$-group can be assigned a Roquette $p$-group, called the genotype. This has already been done by Bouc and Kronstein in the special cases Q and C. A genetic invariant of an irrep is invariant under group isomorphism, change of coefficient field, Galois conjugation, and under suitable inductions from subquotients. It turns out that the genetic invariants are precisely the invariants of the genotype. We shall examine relationships between some genetic invariants and the genotype. As an application, we shall count Galois conjugacy classes of certain kinds of irreps.