Parabolic-equivariant modules over polynomial rings in infinitely many variables

Abstract

We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}({\mathbf{C}}^{\infty })$ consisting of elements fixing a flag in ${\mathbf{C}}^{\infty }$ with each graded piece infinite-dimensional. We decompose the category into simpler pieces that can be described combinatorially, and prove a number of finiteness results, such as finite generation of local cohomology and rationality of Hilbert series. Furthermore, we show that this category is equivalent to the category of representations of a particular combinatorial category generalizing $\mathbf{FI}$.

26 pages, comments welcome