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Ideal transition systems

Eric Marberg, Brendan Pawlowski

TL;DR

This work develops an inductive framework, called transition systems, to compute initial ideals and Gröbner bases for families of ideals arising from matrix Schubert varieties and their skew-symmetric analogues. By formulating systems of equalities from transition recurrences, the authors recover canonical Gröbner bases and primality results in the classical and skew-symmetric settings, and they illuminate obstructions in the symmetric case, proposing extended structures like transition forests. The approach unifies orbit-closure geometry with combinatorial indexing to produce finite, verifiable transition systems for key families, and it links these algebraic constructions to stable limits in equivariant K-theory and orthogonal Grothendieck polynomials. The work also highlights open conjectures about symmetric matrix Schubert varieties, outlining evidence and potential consequences if such a transition system exists. Together, these results provide a cohesive, inductive toolkit for understanding determinantal ideals associated with Schubert-type varieties and their stable invariants.

Abstract

We study an inductive method of computing initial ideals and Gröbner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs $(I,J)$ where $I$ is any ideal and $J$ is a monomial ideal contained in the initial ideal of $I$. These containments become a system of equalities if one can establish a particular transition recurrence among the chosen ideals. We describe explicit constructions of such systems in two motivating cases -- namely, for the ideals of matrix Schubert varieties and their skew-symmetric analogues. Despite many formal similarities with these examples, for the symmetric versions of matrix Schubert varieties, it is an open problem to construct the same kind of transition system. We present several conjectures that would follow from such a construction, while also discussing the special obstructions arising in the symmetric case.

Ideal transition systems

TL;DR

This work develops an inductive framework, called transition systems, to compute initial ideals and Gröbner bases for families of ideals arising from matrix Schubert varieties and their skew-symmetric analogues. By formulating systems of equalities from transition recurrences, the authors recover canonical Gröbner bases and primality results in the classical and skew-symmetric settings, and they illuminate obstructions in the symmetric case, proposing extended structures like transition forests. The approach unifies orbit-closure geometry with combinatorial indexing to produce finite, verifiable transition systems for key families, and it links these algebraic constructions to stable limits in equivariant K-theory and orthogonal Grothendieck polynomials. The work also highlights open conjectures about symmetric matrix Schubert varieties, outlining evidence and potential consequences if such a transition system exists. Together, these results provide a cohesive, inductive toolkit for understanding determinantal ideals associated with Schubert-type varieties and their stable invariants.

Abstract

We study an inductive method of computing initial ideals and Gröbner bases for families of ideals in a polynomial ring. This method starts from a given set of pairs where is any ideal and is a monomial ideal contained in the initial ideal of . These containments become a system of equalities if one can establish a particular transition recurrence among the chosen ideals. We describe explicit constructions of such systems in two motivating cases -- namely, for the ideals of matrix Schubert varieties and their skew-symmetric analogues. Despite many formal similarities with these examples, for the symmetric versions of matrix Schubert varieties, it is an open problem to construct the same kind of transition system. We present several conjectures that would follow from such a construction, while also discussing the special obstructions arising in the symmetric case.

Paper Structure

This paper contains 23 sections, 33 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gröbner bases
  4. More on initial ideals
  5. Transition systems
  6. Main definition
  7. Examples
  8. Orbit closures
  9. Reductions
  10. Matrix Schubert varieties
  11. A reverse lexicographic transition system
  12. Finite indexing set
  13. Proof of Theorem \ref{['msv-thm']}
  14. Skew-symmetric matrix Schubert varieties
  15. Another reverse lexicographic transition system
  16. ...and 8 more sections

Key Result

Proposition 2.7

Suppose $I$ and $J$ are ideals in $\mathbb{K}[{\bf x}]$. Then:

Figures (2)

  • Figure 1: Ideals of $\mathbb{K}[\mathsf{Mat}^{\mathrm{sym}}_{3\times 3}]$ defined for the transition system $\mathcal{T}_\mathscr{I}$ in Example \ref{['sym3-ex']}
  • Figure 2: Transition forest constructed for Example \ref{['sym3-ex']}. The boxed ideals $\boxed{I}$ are the ideals not of the form $I(\mathsf{MSV}^{\mathrm{sym}}_w)$ for any $w \in \mathsf{Mat}^{\mathrm{sym}}_{3\times 3}$.

Theorems & Definitions (98)

  • Example 1.1
  • Example 1.2
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Proposition 2.7: See MP2022
  • Corollary 2.8
  • ...and 88 more