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Matrix Hessenberg schemes over the minimal sheet

Rebecca Goldin, Martha Precup

TL;DR

This work studies flat degenerations of matrix Hessenberg schemes over sheets in the Lie algebra \\mathfrak{gl}_n(\\mathbb{C}). It constructs a sheet-line \\mathsf{x}_t inside the minimal sheet and proves that the one-parameter family \\mathcal{Y}_{\mathsf{x}_t,h} \\to \\operatorname{Spec} \\mathbb{C}[t] has general fiber \\mathcal{Y}_{\mathsf{x},h} and special fiber \\mathcal{Y}_{\mathsf{n},h}, with flatness established for the minimal sheet (where \\mathsf{n} is the minimal nilpotent). The paper also analyzes the associated primes, proves flatness extends to all fibers in the minimal sheet, and shows that if the Hessenberg function h is indecomposable, the nilpotent fiber is reduced and decomposes into matrix Schubert varieties; conversely, for decomposable h, nonreduced behavior and embedded components can occur. Beyond the minimal sheet, the authors propose a conjecture that similar flatness holds over other sheets and develop a robust Gröbner-basis/saturation framework to study these degenerations. These results yield consequences for dimensions and cohomology classes in flat families and provide a structured degeneration mechanism between semisimple and nilpotent matrix Hessenberg schemes. The methodology combines explicit ideal descriptions, saturation, Gröbner basis techniques, and a detailed analysis of associated primes to control geometric features across fibers.

Abstract

We study families of matrix Hessenberg schemes in the affine scheme of complex $n\times n$ matrices, each defined over a fixed sheet in the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$. It is well known that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case. For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in $\mathfrak{gl}_n(\mathbb{C})$, this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.

Matrix Hessenberg schemes over the minimal sheet

TL;DR

This work studies flat degenerations of matrix Hessenberg schemes over sheets in the Lie algebra \\mathfrak{gl}_n(\\mathbb{C}). It constructs a sheet-line \\mathsf{x}_t inside the minimal sheet and proves that the one-parameter family \\mathcal{Y}_{\mathsf{x}_t,h} \\to \\operatorname{Spec} \\mathbb{C}[t] has general fiber \\mathcal{Y}_{\mathsf{x},h} and special fiber \\mathcal{Y}_{\mathsf{n},h}, with flatness established for the minimal sheet (where \\mathsf{n} is the minimal nilpotent). The paper also analyzes the associated primes, proves flatness extends to all fibers in the minimal sheet, and shows that if the Hessenberg function h is indecomposable, the nilpotent fiber is reduced and decomposes into matrix Schubert varieties; conversely, for decomposable h, nonreduced behavior and embedded components can occur. Beyond the minimal sheet, the authors propose a conjecture that similar flatness holds over other sheets and develop a robust Gröbner-basis/saturation framework to study these degenerations. These results yield consequences for dimensions and cohomology classes in flat families and provide a structured degeneration mechanism between semisimple and nilpotent matrix Hessenberg schemes. The methodology combines explicit ideal descriptions, saturation, Gröbner basis techniques, and a detailed analysis of associated primes to control geometric features across fibers.

Abstract

We study families of matrix Hessenberg schemes in the affine scheme of complex matrices, each defined over a fixed sheet in the Lie algebra . It is well known that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case. For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in , this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.
Paper Structure (21 sections, 44 theorems, 124 equations)

This paper contains 21 sections, 44 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Relevant matrix schemes
  4. Matrix Schubert varieties
  5. Matrix Hessenberg schemes
  6. Sheets and Families of matrix Hessenberg schemes
  7. Sheets
  8. Families of matrix Hessenberg schemes over arbitrary sheets
  9. Flat morphisms and miracle flatness
  10. Flat morphisms
  11. Miracle flatness and applications
  12. Technical lemmas
  13. The ideals $\mathcal{J}^t_i$ and $\mathcal{K}_j$
  14. Gröbner background
  15. Saturation proofs
  16. ...and 6 more sections

Key Result

Theorem 1

Let $h: \{1,\ldots, n\}\to \{1,\ldots, n\}$ be a Hessenberg function. For all $\mathsf{x}\in {\mathfrak g}_{(2,1^{n-2})}$ there exists a curve $\mathsf{x}_t$ in the minimal sheet ${\mathfrak g}_{(2,1^{n-2})}$ and a flat morphism with general fiber $\mathcal{Y}_{\mathsf{x}, h}$ and special fiber $\mathcal{Y}_{\mathsf{n},h}$, where $\mathsf{n}$ is an element of the minimal nilpotent orbit in $\math

Theorems & Definitions (92)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 82 more