Constructing equivariant vector bundles via the BGG correspondence

TL;DR

This paper presents a method to construct and identify equivariant vector bundles on projective space using the BGG correspondence, with a focus on bundles with symmetry under the alternating group on five points.

Contribution

It introduces a new strategy for constructing finitely generated G-equivariant modules and provides a criterion to identify when these modules correspond to vector bundles.

Findings

Developed a necessary condition for modules to correspond to vector bundles.

Applied the method to classify strongly determined equivariant vector bundles on P^4.

Enabled explicit construction of equivariant vector bundles with group symmetry.

Abstract

We describe a strategy for the construction of finitely generated $G$-equivariant $\mathbb{Z}$-graded modules $M$ over the exterior algebra for a finite group $G$. By an equivariant version of the BGG correspondence, $M$ defines an object $\mathcal{F}$ in the bounded derived category of $G$-equivariant coherent sheaves on projective space. We develop a necessary condition for $\mathcal{F}$ being isomorphic to a vector bundle that can be simply read off from the Hilbert series of $M$. Combining this necessary condition with the computation of finite excerpts of the cohomology table of $\mathcal{F}$ makes it possible to enlist a class of equivariant vector bundles on $\mathbb{P}^4$ that we call strongly determined in the case where $G$ is the alternating group on $5$ points.