Elementary Equivariant Modules

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.