# ALGEBRAIC COMBINATORICS

Toric degenerations of Grassmannians from matching fields
Algebraic Combinatorics, Volume 2 (2019) no. 6, p. 1109-1124

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 $\text{Gr}\left(3,n\right)$.

Revised : 2019-03-04
Accepted : 2019-03-21
Published online : 2019-12-04
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},
publisher = {MathOA foundation},
volume = {2},
number = {6},
year = {2019},
pages = {1109-1124},
doi = {10.5802/alco.77},
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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