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 $\mathbf{ FI}$.
Categories: math.AC math.RT