GL-equivariant modules over polynomial rings in infinitely many variables

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.Comment: 59 pages, uses ytableau.sty; v2: expanded details in many proofs especially in Sections 2 and 4, Section 6 substantially expanded, added references; v3: corrected typos and Remark 4.3.3 from published versio

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized