English

The Hilbert series of the superspace coinvariant ring

Combinatorics 2024-11-20 v2

Abstract

Let Ωn\Omega_n be the ring of polynomial-valued holomorphic differential forms on complex nn-space, referred to in physics as the superspace ring of rank nn. The symmetric group Sn\mathfrak{S}_n acts diagonally on Ωn\Omega_n by permuting commuting and anticommuting generators simultaneously. We let SInΩnSI_n \subseteq \Omega_n be the ideal generated by Sn\mathfrak{S}_n-invariants with vanishing constant term and study the quotient SRn=Ωn/SInSR_n = \Omega_n / SI_n of superspace by this ideal. We calculate the doubly-graded Hilbert series of SRnSR_n and prove an `operator theorem' which characterizes the harmonic space SHnΩnSH_n \subseteq \Omega_n attached to SRnSR_n in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results which were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Li, Machacek, Sulzgruber, Swanson, Wallach, and Zabrocki.

Keywords

Cite

@article{arxiv.2301.09763,
  title  = {The Hilbert series of the superspace coinvariant ring},
  author = {Brendon Rhoades and Andy Wilson},
  journal= {arXiv preprint arXiv:2301.09763},
  year   = {2024}
}

Comments

32 pages, 1 figure