We study the spaces Q_m of m-quasi-invariant polynomials of the symmetric group S_n in characteristic p. Using the representation theory of the symmetric group we describe the Hilbert series of Q_m 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 Q_m 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 Q_m has a different Hilbert series over characteristic 0 and characteristic p, and what the degrees of the generators of Q_m are in such cases. We also extend various results to the spaces Q_m,q of q-deformed m-quasi-invariants and prove a sufficient condition for the Hilbert series of Q_m,q to differ from the Hilbert series of Q_m.

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.