Neutral representations in dimension $\leq 3$ and fields of moduli

TL;DR

This paper classifies neutral representations of algebraic groups in low dimensions, provides criteria for their neutrality in abelian cases, and introduces the concept of the normalizer of gerbe morphisms, with applications to fields of moduli.

Contribution

It offers a complete classification of neutral, faithful representations of finite groups in dimension ≤ 3, and develops a general framework for analyzing the normalizer of gerbe morphisms.

Findings

Classified all neutral, faithful finite group representations in dimension ≤ 3.

Provided a computation-friendly criterion for neutrality of finite abelian group representations.

Introduced the concept of the normalizer of gerbe morphisms, depending only on geometric type.

Abstract

A representation $V$ of an algebraic group $G$ induces a vector bundle $[V/G] \to BG$. The representation $V$ of $G$ is neutral if, for every twisted form $\mathcal{V} \to \mathcal{G}$ of $[V/G] \to BG$ over a field $k$, we have $\mathcal{G}(k) \neq \emptyset$. Twisted forms of representations arise in many ways, for instance as cohomology of families of varieties on residual gerbes of moduli spaces, and from quotient singularities. Moreover, every Tannakian category is the category of vector bundles on some gerbe. Because of this, studying neutral representations yields numerous applications, especially to problems about fields of moduli. The present article has three main results. First, we completely classify neutral, faithful representations of finite groups in dimension $\leq 3$. Second, we give a very general, computation-friendly result for proving that representations of finite abelian groups are neutral, in arbitrary dimensions. Third, we develop the abstract concept of the normalizer $\mathcal{G} \to \mathcal{N} \to \mathcal{H}$ of a morphism of gerbes $\mathcal{G} \to \mathcal{H}$ on an arbitrary site (twisted representations correspond to morphisms of gerbes $\mathcal{G} \to B\mathrm{GL}_{n}$), and show that the normalizer $\mathcal{N}$ only depends on the geometric type of $\mathcal{G} \to \mathcal{H}$.