New York Journal of Mathematics
Volume 29 (2023), 613-634


Frank Wang

Toward explicit Hilbert series of quasi-invariant polynomials in characteristic p and q-deformed quasi-invariants

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Published: May 13, 2023.
Keywords: Quasi-invariant polynomials, q-deformations, Opdam shift operators.
Subject [2020]: 81R12 (primary), 13A50, 20C08 (secondary).

We study the spaces Qm of m-quasi-invariant polynomials of the symmetric group Sn in characteristic p. Using the representation theory of the symmetric group we describe the Hilbert series of Qm for n=3, proving a conjecture of Ren and Xu [12]. From this we may deduce the palindromicity and highest term of the Hilbert polynomial and the freeness of Qm as a module over the ring of symmetric polynomials, which are conjectured for general n. We also prove further results in the case n=3 that allow us to compute values of m,p for which Qm has a different Hilbert series over characteristic 0 and characteristic p, and what the degrees of the generators of Qm are in such cases. We also extend various results to the spaces Qm,q of q-deformed m-quasi-invariants and prove a sufficient condition for the Hilbert series of Qm,q to differ from the Hilbert series of Qm.


I would like to thank Calder Morton-Ferguson for mentoring this project and for providing me with various resources, advice, and ideas. I would like to thank Roman Bezrukavnikov for suggesting the original project and for useful discussions. I would also like to thank Pavel Etingof for suggesting a closely related but more tractable problem to pursue and for providing resources and valuable insight to help tackle the problem. I would also like to thank Michael Ren for useful discussions regarding his paper with Xu [12]. I would like to thank Ankur Moitra and David Jerison for overall advice about the project and research in general. Lastly, I would like to thank Slava Gerovitch and the MIT Math Department for running the Summer Program for Undergraduate Research (SPUR) and Undergraduate Research Opportunities Program (UROP), during which this research was conducted.

Author information

Frank Wang
Department of Mathematics
77 Massachusetts Ave
Cambridge, MA 02139, USA