Noetherianity of some degree two twisted skew-commutative algebras
- 4 Downloads
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 \(\hbox {Sym}(\hbox {Sym}^2(\mathbf {C}^{\infty }))\) and \(\hbox {Sym}(\bigwedge ^2(\mathbf {C}^{\infty }))\). In this paper, we establish noetherianity for the skew-commutative versions of these two algebras, namely \(\bigwedge (\hbox {Sym}^2(\mathbf {C}^{\infty }))\) and \(\bigwedge (\bigwedge ^2(\mathbf {C}^{\infty }))\). 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.
Mathematics Subject Classification
13E05 13A50Notes
References
- 1.Cheng, S.J., Wang, W.: Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, vol. 144. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
- 2.Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015). arXiv:1204.4533v4 MathSciNetCrossRefGoogle Scholar
- 3.Fulton, W.: Young Tableaux, with Applications to Representation Theory and Geometry, London Mathematical Society Student Texts 35. Cambridge University Press, Cambridge (1997)Google Scholar
- 4.Lam, T.Y.: A first Course in Noncommutative Rings, Graduate Texts in Mathematics 131, 2nd edn. Springer, New York (2001)CrossRefGoogle Scholar
- 5.Macdonald, I.G.: Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1995)Google Scholar
- 6.Miller, J., Wilson, J.: Higher order representation stability and ordered configuration spaces of manifolds (2016). arXiv:1611.01920v1
- 7.Nagpal, R., Sam, S.V., Snowden, A.: Noetherianity of some degree two twisted commutative algebras. Selecta Math. (N.S.) 22(2), 913–937 (2016). arXiv:1501.06925v2 MathSciNetCrossRefGoogle Scholar
- 8.Raicu, C., Weyman, J.: The syzygies of some thickenings of determinantal varieties. Proc. Am. Math. Soc. 145(1), 49–59 (2017). arXiv:1411.0151v2 MathSciNetCrossRefGoogle Scholar
- 9.Sam, S., Snowden, A.: GL-equivariant modules over polynomial rings in infinitely many variables. Trans. Am. Math. Soc. 368, 1097–1158 (2016). arXiv:1206.2233v3 MathSciNetCrossRefGoogle Scholar
- 10.Sam, S., Snowden, A.: Introduction to twisted commutative algebras. arXiv:1209.5122v1
- 11.Sam, S.V., Snowden, A.: Stability patterns in representation theory, Forum. Math. Sigma 3, e11, 108 pp. (2015). arXiv:1302.5859v2
- 12.Serganova, V.: Classical Lie Superalgebras at Infinity. In: Gorelik, M., Papi, P. (eds.) Advances in Lie Superalgebras. Springer INdAM Series, vol. 7, pp. 181–201. Springer, New York (2014)CrossRefGoogle Scholar