• Corpus ID: 246016442

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

@inproceedings{Wang2022TowardEH,
  title={Toward explicit Hilbert series of quasi-invariant polynomials in characteristic \$p\$ and \$q\$-deformed quasi-invariants},
  author={Frank Wang},
  year={2022}
}

• Frank Wang
• Published 16 January 2022
• Mathematics

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 [10]. 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.