Sp-equivariant modules over polynomial rings in infinitely many variables

Abstract

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M fits into an exact triangle $T\to M\to F\to$ where T is a finite length complex of torsion modules and F is a finite length complex of "free" modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras $\mathrm{Sym}({\mathbf{C}}^{\infty }\oplus {\bigwedge }^2{\mathbf{C}}^{\infty })$ and $\mathrm{Sym}({\mathbf{C}}^{\infty }\oplus {\mathrm{Sym}}^2{\mathbf{C}}^{\infty })$ are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian.

29 pages

References in corpus (1)

• The representation theory of Brauer categories I: triangular categories

Cited by in corpus (5)

• The representation theory of Brauer categories I: triangular categories
• On the geometry and representation theory of isomeric matrices
• GL-algebras in positive characteristic I: the exterior algebra
• Sym-Noetherianity for powers of GL-varieties
• A Tensor Restriction Theorem over Finite Fields