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New kinds of block diagonal matching fields and toric degenerations of Grassmannians

Akihiro Higashitani, Nobukazu Kowaki

TL;DR

The work advances the study of toric degenerations of Grassmannians by introducing $({\mathbf a},\ell)$-block diagonal matching fields, a broad generalization of prior block diagonal constructions. It provides crisp SAGBI-basis criteria: for $\ell\ge 4$, a SAGBI basis (and hence toric degeneration) holds when $a_1\le 3$ and all $a_i\ (2\le i\le s-1)\le 2$, while it fails under certain larger first blocks or sum constraints, with the $\ell=3$ case universally giving a SAGBI basis for all ${\mathbf a}$. The results connect coherent matching fields to toric degenerations via the Plücker algebra and its tableaux encoding, and they identify both constructive cases and obstructions. This yields a new, explicit family of toric degenerations of $Gr(r,n)$ with potential implications for Newton polytopes and polyhedral geometry of Grassmannians. The paper also analyzes new initial monomials arising from vertical tableau swaps, clarifying how leading terms interact under the $({\mathbf a},\ell)$-block diagonal framework.

Abstract

Block diagonal matching field has many previous works. In general, a coherent matching field induces a monomial order to Plücker algebra, and block diagonal matching fields are a kind of coherent matching fields. In the present paper, we introduce a new kind of block diagonal matching fields and study the problem when they give a SAGBI basis. As a corollary, we provide a new family of toric degenerations of Grassmannians by using SAGBI bases.

New kinds of block diagonal matching fields and toric degenerations of Grassmannians

TL;DR

The work advances the study of toric degenerations of Grassmannians by introducing -block diagonal matching fields, a broad generalization of prior block diagonal constructions. It provides crisp SAGBI-basis criteria: for , a SAGBI basis (and hence toric degeneration) holds when and all , while it fails under certain larger first blocks or sum constraints, with the case universally giving a SAGBI basis for all . The results connect coherent matching fields to toric degenerations via the Plücker algebra and its tableaux encoding, and they identify both constructive cases and obstructions. This yields a new, explicit family of toric degenerations of with potential implications for Newton polytopes and polyhedral geometry of Grassmannians. The paper also analyzes new initial monomials arising from vertical tableau swaps, clarifying how leading terms interact under the -block diagonal framework.

Abstract

Block diagonal matching field has many previous works. In general, a coherent matching field induces a monomial order to Plücker algebra, and block diagonal matching fields are a kind of coherent matching fields. In the present paper, we introduce a new kind of block diagonal matching fields and study the problem when they give a SAGBI basis. As a corollary, we provide a new family of toric degenerations of Grassmannians by using SAGBI bases.
Paper Structure (14 sections, 8 theorems, 28 equations)

This paper contains 14 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Plücker algebra and SAGBI basis
  4. Matching fields
  5. Matching field tableaux
  6. $({\mathbf a},\ell)$-block diagonal matching fields
  7. Definition of $({\mathbf a},\ell)$-block diagonal matching fields
  8. Properties on $\Lambda_{{\mathbf a},\ell}$
  9. New initial monomials
  10. $({\mathbf a},\ell)$-block diagonal matching fields which do not provide a SAGBI basis
  11. Toric degenerations associated to $({\mathbf a},\ell)$-block diagonal matching fields
  12. "The first" part of Theorem \ref{['thm:main']}
  13. Proof of the case $\ell=3$
  14. Remaining cases

Key Result

Theorem 1.1

Let ${\mathbf a}=(a_1,\ldots,a_s) \in {\mathbb Z}_{>0}^s$ satisfying $\sum_{i=1}^s a_i=n$ and $a_s \geq 2$, and let $\ell \geq 4$. Consider the $({\mathbf a},\ell)$-block diagonal matching field $\Lambda_{{\mathbf a},\ell}$. Then the generating set $\{\det({\mathbf x}_I) : I \in {\mathbf I}_{r,n}\}$ On the other hand, the generating set $\{\det({\mathbf x}_I) : I \in {\mathbf I}_{r,n}\}$ of the Pl

Theorems & Definitions (32)

  • Theorem 1.1: See Theorems \ref{['thm:onlyif']}, \ref{['thm:if']} and Lemma \ref{['cor:if']}
  • Theorem 1.2
  • Definition 2.1: SAGBI bases for the Plücker algebra
  • Definition 2.2: Coherent matching fields (sturmfels1993maximal, mohammadi2019toric)
  • Theorem 2.3: sturmfels1996grobner
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 3.1: $({\mathbf a},\ell)$-block diagonal matching fields
  • Proposition 3.3
  • ...and 22 more