Representations of elementary abelian p-groups and bundles on Grassmannians

TL;DR

This paper explores the representation theory of elementary abelian p-groups through geometric invariants and modules that lead to algebraic vector bundles on Grassmannians, expanding understanding of their structure and applications.

Contribution

It introduces new geometric invariants and modules of constant radical and socle type, generalizing existing concepts and connecting representation theory with algebraic geometry.

Findings

Modules of constant radical and socle type produce algebraic vector bundles on Grassmannians.

New invariants based on restrictions to truncated polynomial subalgebras are developed.

Several explicit examples illustrating the theory are provided.

Abstract

We initiate the study of representations of elementary abelian $p$-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by $r$ nilpotent elements, $k[t_1,..., t_r]/(t^p_1,..., t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.