The Geometry of Polynomial Representations

We define a GL-variety to be an (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank and played a key role in two recent proofs of Stillman’s conjecture. We initiate a systematic study of GL-varieties and establish a number of foundational results about them. For example, we prove a version of Chevalley’s theorem on constructible sets in this setting.