The geometry of polynomial representations

Abstract

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.

46 pages

References in corpus (4)

• FI-modules and the cohomology of modular representations of symmetric groups
• Universality of high-strength tensors
• The set of forms with bounded strength is not closed
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

Cited by in corpus (9)

• Universality of high-strength tensors
• The set of forms with bounded strength is not closed
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
• GL-algebras in positive characteristic I: the exterior algebra
• Stable representation theory: beyond the classical groups
• Dual Infinite Wedge is ${\mathrm{GL}}_{\infty }$-equivariantly noetherian
• Sym-Noetherianity for powers of GL-varieties
• A Tensor Restriction Theorem over Finite Fields
• Topological Noetherianity of the infinite half-spin representations