Small modules with interesting rank varieties

TL;DR

This paper investigates the properties of rank varieties for modules over group algebras of elementary abelian p-groups, providing conditions for Green vertices and exploring the construction of modules with prescribed rank varieties.

Contribution

It offers a sufficient condition for Green vertices based on rank varieties and examines the construction of small modules with specific rank varieties, including a detailed case for symmetric groups.

Findings

Established a criterion linking Green vertices to rank varieties.

Explored the existence of small modules with given algebraic varieties as rank varieties.

Analyzed a specific simple module for symmetric groups.

Abstract

This paper focuses on the rank varieties for modules over a group algebra $\mathbb{F}E$ where $E$ is an elementary abelian $p$-group and $p$ is the characteristic of an algebraically closed field $\mathbb{F}$. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module containing an elementary abelian $p$-group $E$ in terms of the rank variety of the module restricted to $E$. In the second part, given a homogeneous algebraic variety $V$ , we explore the problem on finding a small module with rank variety $V$ . In particular, we examine the simple module $D^{(kp-p+1,1^{p-1})}$ for the symmetric group $\mathfrak{S}_{kp}$.