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 $\textbf {GL}$-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.

中文翻译：

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多项式表示的几何

我们将 GL 变体定义为（通常为无限维）代数变体，具有无限一般线性群的作用，在该群下，坐标环形成多项式表示。这些变体已被用于研究强度和张量等级等不变量的渐近性质，并在最近的两个斯蒂尔曼猜想的证明中发挥了关键作用。我们对 $\textbf {GL}$-varieties 进行了系统研究，并建立了一些关于它们的基础结果。例如，我们在这种情况下证明了 Chevalley 定理关于可构造集合的一个版本。