Toric degenerations of Grassmannians from matching fields

TL;DR

This paper explores how matching fields can be used to generate toric degenerations of Grassmannians, providing conditions for their algebraic properties and introducing new families of degenerations.

Contribution

It establishes a necessary and sufficient condition for matching fields to produce Khovanskii bases and introduces 2-block diagonal matching fields for Grassmannian degenerations.

Findings

Necessary condition for matching fields to yield Khovanskii bases.

Characterization of quadratically generated ideals for matching fields.

Introduction of 2-block diagonal matching fields leading to new toric degenerations.

Abstract

We study the algebraic combinatorics of monomial degenerations of Pl\"ucker forms which is governed by matching fields in the sense of Sturmfels and Zelevinsky. We provide a necessary condition for a matching field to yield a Khovanskii basis of the Pl\"ucker algebra for $3$-planes in $n$-space. When the ideal associated to the matching field is quadratically generated this condition is both necessary and sufficient. Finally, we describe a family of matching fields, called $2$-block diagonal, whose ideals are quadratically generated. These matching fields produce a new family of toric degenerations of $\Gr(3, n)$.