Elementary Equivariant Modules

TL;DR

This paper introduces elementary equivariant modules over polynomial rings with GL(V) symmetry, showing they form the building blocks for all finitely generated equivariant modules and detailing their resolutions.

Contribution

It defines elementary equivariant modules $M_{}$ and proves they form a filtration basis for all finitely generated equivariant modules, including their resolutions.

Findings

Any finitely generated equivariant module admits a filtration with modules of two types.

Each elementary equivariant module $M_{}$ has a linear resolution.

Resolutions of truncations of $M_{}$ are also described.

Abstract

We study equivariant modules over $GL(V)$ over the polynomial ring $R = Sym V$. We introduce for every partition $\lambda$ the elementary equivariant module $M_{\lambda}$. Then we prove that any finitely generated equivariant module admits a fltration with associated graded being the direct sum of modules of only two kinds: either $M_{\lambda}$ or truncations of $M_{\lambda}$. We show that each $M_{\lambda}$ has a linear resolution and describe also the resolution of its truncations.