Toric degenerations of Grassmannians from matching fields

Mohammadi, Fatemeh; Shaw, Kristin

doi:10.5802/alco.77

Mohammadi, Fatemeh; Shaw, Kristin

Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1109-1124.

Abstract

We study the algebraic combinatorics of monomial degenerations of Plücker 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 SAGBI basis of the Plücker 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 $Gr (3, n)$ .

Received: 2018-09-18
Revised: 2019-03-04
Accepted: 2019-03-21
Published online: 2019-12-04

MR 4049839 | Zbl 07140426

DOI: https://doi.org/10.5802/alco.77
Classification: 14M15, 14M25, 14T05
Keywords: toric degenerations, SAGBI and Khovanskii bases, Grassmannians, tropical geometry

@article{ALCO_2019__2_6_1109_0,
     author = {Mohammadi, Fatemeh and Shaw, Kristin},
     title = {Toric degenerations of Grassmannians from matching fields},
     journal = {Algebraic Combinatorics},
     pages = {1109--1124},
     publisher = {MathOA foundation},
     volume = {2},
     number = {6},
     year = {2019},
     doi = {10.5802/alco.77},
     mrnumber = {4049839},
     zbl = {07140426},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1109_0/}
}

Mohammadi, Fatemeh; Shaw, Kristin. Toric degenerations of Grassmannians from matching fields. Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 1109-1124. doi : 10.5802/alco.77. https://alco.centre-mersenne.org/item/ALCO_2019__2_6_1109_0/

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