GL-equivariant modules over polynomial rings in infinitely many variables

Abstract

Abstract. Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly.Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

Stability Patterns in Representation Theory

“…However, most of the real work on this -computing the derived functors of the specialization functor on simple objects -was done elsewhere ( [SSW] for the classical groups and [SS1] for the symmetric group). In this paper, we develop the basic theory of the specialization functor, and explain how the cited results (which use a different language) can be rephrased using it.…”

“…It follows from our results that each of the categories under consideration is Koszul. The category Rep pol (GA) ∼ = Rep(S), is Koszul self-dual: this was established in [SS1], where we constructed a canonical auto-equivalence of the derived category, called the Fourier transform, realizing the auto-duality. Here, we extend this construction to Rep(GL), showing that it is its own dual.…”

“…For symmetric groups, the existence of such a ring follows from Murnaghan's theorem (which was first completely proved in [Lit2]). As far as we are aware, the specialization map in this context was not studied until [SS1], and even there it is phrased in a different language. These stable character rings can be viewed as the Grothendieck groups of the categories we study.…”

“…However, to the best of our knowledge, they were first treated from the perspective of commutative algebra in [Sno]. We have since developed the theory further in [SS1,SS2]. One of our motivations for studying the representation theory of O(∞) was to understand O(∞)-analogs of tcas.…”

“…equipped with its natural GL(∞) action. We gave a detailed analysis of the category of modules over this tca in [SS1]. This category was also studied in [CEF], where such modules are called FI-modules.…”