On Quasiinvariant Polynomials

Michael Ren Phillips Academy

S.-T. Yau High School Science Awarded Papers mathscidoc:1801.35035

Yau Science Award (Math), 2017.12
Symmetric polynomials are polynomials that are invariant under the action of the symmetric group, and they play an integral role in mathematics. The space of quasiin- variant polynomials, polynomials that are invariant under the action of the symmetric group to a certain order, were introduced by Feigin and Veselov. These spaces are modules over the ring of symmetric polynomials, and their Hilbert series in elds of characteristic 0 were also computed by Feigin and Veselov. In this paper, we study the Hilbert series of these spaces in elds of positive char- acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the space is twisted by a monomial. We extend some of their results to spaces twisted by a product of smooth functions and compute the Hilbert series of the space in certain cases.
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  title={On Quasiinvariant Polynomials},
  author={Michael Ren},
  booktitle={Yau Science Award (Math)},
Michael Ren. On Quasiinvariant Polynomials. 2017. In Yau Science Award (Math). http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180118031212500750910.
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