Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Superspace coinvariants and hyperplane arrangements

Robert Angarone, Patricia Commins, Trevor Karn, Satoshi Murai, Brendon Rhoades

TL;DR

This work resolves the first explicit basis for the superspace coinvariant ring $SR = \Omega / I$ by connecting superspace coinvariants to Solomon–Terao algebras of carefully chosen southwest hyperplane arrangements. The authors establish a transfer principle that reduces the problem to commutative-algebra questions about colon ideals $S^{\mathfrak{S}_n}_+ : f_J$ and their bases, proving that the monomial families ${\mathcal M}(J)$ descend to bases for these quotients and, in turn, assemble into an explicit Artin basis for $SR$. They prove the Sagan–Swanson conjecture by constructing the Artin basis ${\mathcal M}$ for $SR$ and relate $SR$ to a family of ST-algebras arising from southwest arrangements, which are themselves free and possess complete-intersection quotients with Hilbert-series governed by the exponents ${\mathrm{st}}(J)$. The approach bridges superspace coinvariants with Hessenberg-type geometry via the Solomon–Terao framework, offering a robust method to obtain explicit bases and suggesting avenues for generalization to other types and geometric interpretations. Overall, the paper delivers the first explicit basis for $SR$, proves key conjectures, and highlights deep connections between hyperplane arrangements, invariant theory, and combinatorial algebra.

Abstract

Let $Ω$ be the {\em superspace ring} of polynomial-valued differential forms on affine $n$-space. The natural action of the symmetric group $\mathfrak{S}_n$ on $n$-space induces an action of $\mathfrak{S}_n$ on $Ω$. The {\em superspace coinvariant ring} is the quotient $SR$ of $Ω$ by the ideal generated by $\mathfrak{S}_n$-invariants with vanishing constant term. We give the first explicit basis of $SR$, proving a conjecture of Sagan and Swanson. Our techniques use the theory of hyperplane arrangements. We relate $SR$ to instances of the Solomon-Terao algebras of Abe-Maeno-Murai-Numata and use exact sequences relating the derivation modules of certain `southwest closed' arrangements to obtain the desired basis of $SR$.

Superspace coinvariants and hyperplane arrangements

TL;DR

This work resolves the first explicit basis for the superspace coinvariant ring by connecting superspace coinvariants to Solomon–Terao algebras of carefully chosen southwest hyperplane arrangements. The authors establish a transfer principle that reduces the problem to commutative-algebra questions about colon ideals and their bases, proving that the monomial families descend to bases for these quotients and, in turn, assemble into an explicit Artin basis for . They prove the Sagan–Swanson conjecture by constructing the Artin basis for and relate to a family of ST-algebras arising from southwest arrangements, which are themselves free and possess complete-intersection quotients with Hilbert-series governed by the exponents . The approach bridges superspace coinvariants with Hessenberg-type geometry via the Solomon–Terao framework, offering a robust method to obtain explicit bases and suggesting avenues for generalization to other types and geometric interpretations. Overall, the paper delivers the first explicit basis for , proves key conjectures, and highlights deep connections between hyperplane arrangements, invariant theory, and combinatorial algebra.

Abstract

Let be the {\em superspace ring} of polynomial-valued differential forms on affine -space. The natural action of the symmetric group on -space induces an action of on . The {\em superspace coinvariant ring} is the quotient of by the ideal generated by -invariants with vanishing constant term. We give the first explicit basis of , proving a conjecture of Sagan and Swanson. Our techniques use the theory of hyperplane arrangements. We relate to instances of the Solomon-Terao algebras of Abe-Maeno-Murai-Numata and use exact sequences relating the derivation modules of certain `southwest closed' arrangements to obtain the desired basis of .
Paper Structure (18 sections, 23 theorems, 98 equations)

This paper contains 18 sections, 23 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Commutative algebra
  4. Arrangement combinatorics
  5. Derivation modules
  6. Solomon--Terao algebras
  7. Southwest arrangements
  8. Southwest combinatorics
  9. Southwest algebra
  10. Free bases for southwest arrangements
  11. Exact sequences of derivation modules
  12. The algebra $S/((S_+^{\mathfrak S_n}):f_J)$ and the arrangement ${\mathcal{A}}_J$
  13. Monomial bases of Solomon--Terao algebras
  14. Artin basis of $SR$
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 1.2

(Rhoades--Wilson RW) For each $J \subseteq [n]$, suppose that ${\mathcal{B}}(J) \subseteq S$ is a set of homogeneous polynomials which descends to a basis of $S/( (S^{{\mathfrak{S}}_n}_+) : f_J)$. Then descends to a basis of $SR$.

Theorems & Definitions (47)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 1.3
  • Lemma 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 37 more