Ideal transition systems

TL;DR

This paper introduces an inductive method for computing initial ideals and Gr"obner bases using transition systems, with explicit constructions for matrix Schubert varieties and their skew-symmetric analogues, and discusses open problems in symmetric cases.

Contribution

It provides explicit transition systems for certain classes of ideals and explores open problems and conjectures for symmetric matrix Schubert varieties.

Findings

Constructed transition systems for matrix Schubert varieties.

Extended the method to skew-symmetric analogues.

Identified open problems and conjectures for symmetric cases.

Abstract

We study an inductive method of computing initial ideals and Gr\"obner 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.