Parabolic-equivariant modules over polynomial rings in infinitely many variables

TL;DR

This paper explores the structure of $ extbf{P}$-equivariant modules over infinite-variable polynomial rings, revealing their decomposition, finiteness properties, and equivalence to a combinatorial category, advancing understanding in infinite-dimensional algebra.

Contribution

It introduces a decomposition of the category into combinatorially describable pieces and establishes finiteness and rationality results, along with an equivalence to a generalized $ extbf{FI}$-category.

Findings

Finite generation of local cohomology

Rationality of Hilbert series

Category equivalence to a combinatorial category

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 $\mathbf{FI}$.