Noetherianity of some degree two twisted skew-commutative algebras

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.

arXiv comment22 pages

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• Higher order representation stability and ordered configuration spaces of manifolds
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• The syzygies of some thickenings of determinantal varieties
• Infinite rank spinor and oscillator representations
• Sp-equivariant modules over polynomial rings in infinitely many variables
• Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
• On the geometry and representation theory of isomeric matrices
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• GL-algebras in positive characteristic I: the exterior algebra
• Some generalizations of Schur functors
• Stable representation theory: beyond the classical groups
• A counterexample to a Gröbner approach for noetherianity of the twisted commutative algebra $\mathrm{Sym}({\mathrm{Sym}}^2({\mathbf{C}}^{\infty }))$