Degree of reductivity of a modular representation

TL;DR

This paper investigates the degree of reductivity in modular group representations, providing explicit calculations for various classes and establishing bounds related to cyclic subgroup sizes.

Contribution

It explicitly computes the degree of reductivity for several classes of modular groups and representations, and links this measure to the size of cyclic subgroups in abelian p-groups.

Findings

Explicit formulas for $ ext{delta}(G,V)$ in several classes

Maximal cyclic subgroup size bounds the degree of reductivity

Provides new insights into modular representation invariants

Abstract

For a finite dimensional representation $V$ of a group $G$ over a field $F$, the degree of reductivity $\delta(G,V)$ is the smallest degree $d$ such that every nonzero fixed point $v\in V^{G}\setminus\{0\}$ can be separated from zero by a homogeneous invariant of degree at most $d$. We compute $\delta(G,V)$ explicitly for several classes of modular groups and representations. We also demonstrate that the maximal size of a cyclic subgroup is a sharp lower bound for this number in the case of modular abelian $p$-groups.