Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

TL;DR

This paper derives explicit formulas for the degree of symmetric Grothendieck polynomials and the Castelnuovo-Mumford regularity of Schubert determinantal ideals, providing counterexamples to existing conjectures and proposing new conjectures.

Contribution

It introduces explicit formulas for degrees and regularities related to Grassmannian permutations and Schubert varieties, challenging previous conjectures and suggesting new ones.

Findings

Explicit formula for degree of Grothendieck polynomial

Counterexample to Kummini-Lakshmibai-Sastry-Seshadri conjecture

New conjecture on regularities of Grassmannian Schubert varieties

Abstract

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties.