Combinatorics

Physics

Algebraic variety

Pure mathematics

Action (physics)

Mathematics

Politics

Statistics

Mathematical proof

Algebra over a field

Variety (cybernetics)

Rank (graph theory)

Mathematical analysis

Political science

Affine variety

Geometry

Representation (politics)

Quantum mechanics

Conjecture

Algebraic geometry

Polynomial

Affine transformation

Polynomial ring

Algebraic number

International Mathematics Research Notices

We define a GL-variety to be a (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.

The Geometry of Polynomial Representations