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The Strong Lefschetz Property of Gorenstein Algebras Generated by Relative Invariants

Takahiro Nagaoka, Akihito Wachi

Abstract

We prove the strong Lefschetz property for Artinian Gorenstein algebras generated by the relative invariants of prehomogeneous vector spaces of commutative parabolic type.

The Strong Lefschetz Property of Gorenstein Algebras Generated by Relative Invariants

Abstract

We prove the strong Lefschetz property for Artinian Gorenstein algebras generated by the relative invariants of prehomogeneous vector spaces of commutative parabolic type.
Paper Structure (14 sections, 7 theorems, 25 equations, 1 table)

This paper contains 14 sections, 7 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. Prehomogeneous vector spaces of commutative parabolic type
  3. $({\rm A}_{2n-1}, n)$
  4. $({\rm B}_m, 1)$ and $({\rm D}_m, 1)$
  5. $({\rm C}_n, n)$
  6. $({\rm D}_{2m}, 2m)$
  7. $({\rm E}_7, 7)$
  8. Main theorem
  9. Proof of the main theorem
  10. $\mathop{\mathrm{ad}}\nolimits(\mathfrak{k})$-module structure of $\mathbb{C}[\mathfrak{n}^+]$
  11. $\mathop{\mathrm{ad}}\nolimits(\mathfrak{k})$-module structure of $R/\mathop{\mathrm{Ann}}\nolimits_R(F)$
  12. Generalized Verma modules
  13. Maximal submodules of $M(\mu)$
  14. Proof of the main theorem

Key Result

Theorem 4

Let $\overline{f} \in \mathbb{C}[\mathfrak{n}^-]$ be the basic relative invariant (see Section sec:pv-parab) of $(K, \mathop{\mathrm{Ad}}\nolimits, \mathfrak{n}^-)$ which is a regular prehomogeneous vector space of commutative parabolic type. Set $R = \mathbb{C}[\mathfrak{n}^+]$, and $F = \overline{

Theorems & Definitions (21)

  • Definition 1: Strong Lefschetz property
  • Definition 2: Prehomogeneous vector space
  • Definition 3: Prehomogeneous vector space of commutative parabolic type
  • Theorem 4
  • Example 5: Type (C$_n,n$)
  • Remark 6: The set of Lefschetz elements
  • Definition 7: strongly orthogonal roots
  • Lemma 8: Decomposition of $\mathbb{C}[\mathfrak{n}^+]$ as an $\mathop{\mathrm{ad}}\nolimits(\mathfrak{k})$-module, Schmid MR259164
  • Example 9: Type (C$_n,n$)
  • Proposition 10: Decomposition of $R / \mathop{\mathrm{Ann}}\nolimits_R(F)$ as an $\mathop{\mathrm{ad}}\nolimits(\mathfrak{k})$-module
  • ...and 11 more