Noetherianity of some degree two twisted skew-commutative algebras

TL;DR

This paper proves noetherianity for certain degree two twisted skew-commutative algebras, extending known results and relying on advanced representation theory, with applications in topology and algebraic stability.

Contribution

It establishes the noetherian property for skew-commutative analogues of symmetric and exterior algebras, a significant advancement in the theory of twisted commutative algebras.

Findings

Proves noetherianity of mbda(Sym^2(C^\u2206))) and mbda(mbda^2(C^))

Relies on Serganova's work on periplectic Lie superalgebra representations

Applications in secondary representation stability in cohomology of configuration spaces

Abstract

A major open problem in the theory of twisted commutative algebras (tca's) is proving noetherianity of finitely generated tca's. For bounded tca's this is easy, in the unbounded case, noetherianity is only known for Sym(Sym^2(C^\infty)) and Sym(\wedge^2(C^\infty)). In this paper, we establish noetherianity for the skew-commutative versions of these two algebras, namely \wedge(Sym^2(C^\infty)) and \wedge(\wedge^2(C^\infty)). The result depends on work of Serganova on the representation theory of the infinite periplectic Lie superalgebra, and has found application in the work of Miller-Wilson on "secondary representation stability" in the cohomology of configuration spaces.