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, p. 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
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/

References

[1] An, Byung Hee; Cho, Yunhyung; Kim, Jang Soo On the f-vectors of Gelfand-Tsetlin polytopes, Eur. J. Comb., Volume 67 (2018), pp. 61-77 | Zbl 1378.52010

[2] Bernstein, David; Zelevinsky, Andrei Combinatorics of maximal minors, J. Algebr. Comb., Volume 2 (1993) no. 2, pp. 111-121 | Article | MR 1229427 | Zbl 0794.13021

[3] Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter M. Oriented Matroids, Cambridge University Press, 1999 | Zbl 0944.52006

[4] Bossinger, Lara; Fang, Xin; Fourier, Ghislain; Hering, Milena; Lanini, Martina Toric degenerations of Gr $(2, n)$ and Gr $(3, 6)$ via plabic Graphs, Ann. Comb., Volume 22 (2018) no. 3, pp. 491-512 | Article | MR 3845745 | Zbl 06948249

[5] Bossinger, Lara; Lamboglia, Sara; Mincheva, Kalina; Mohammadi, Fatemeh Computing toric degenerations of flag varieties, Combinatorial Algebraic Geometry, Springer-Verlag New York, 2017, pp. 247-281 | Article | Zbl 1390.14194

[6] Conca, Aldo; Herzog, Jürgen; Valla, Giuseppe Sagbi bases with applications to blow-up algebras, J. Reine Angew. Math., Volume 474 (1996), pp. 113-138 | MR 1390693 | Zbl 0866.13010

[7] Develin, Mike; Sturmfels, Bernd Tropical convexity, Doc. Math., Volume 9 (2004), pp. 1-27 | MR 2054977 | Zbl 1054.52004

[8] Dochtermann, Anton; Mohammadi, Fatemeh Cellular resolutions from mapping cones, J. Comb. Theory, Ser. A, Volume 128 (2014), pp. 180-206 | Article | MR 3265923 | Zbl 1301.05379

[9] Feigin, Evgeny $𝔾_{a}^{M}$ degeneration of flag varieties, Sel. Math., New Ser., Volume 18 (2012) no. 3, pp. 513-537 | Article | MR 2960025 | Zbl 1267.14064

[10] Fink, Alex; Rincón, Felipe Stiefel tropical linear spaces, J. Comb. Theory, Ser. A, Volume 135 (2015), pp. 291-331 | Article | MR 3366480 | Zbl 1321.15044

[11] Herrmann, Sven; Jensen, Anders; Joswig, Michael; Sturmfels, Bernd How to draw tropical planes, Electron. J. Comb., Volume 16 (2009) no. 2, R6, 26 pages | MR 2515769 | Zbl 1195.14080

[12] Herzog, Jürgen; Hibi, Takayuki Distributive lattices, bipartite graphs and Alexander duality, J. Algebr. Comb., Volume 22 (2005) no. 3, pp. 289-302 | Article | MR 2181367 | Zbl 1090.13017

[13] Hibi, Takayuki Every affine graded ring has a Hodge algebra structure, Rend. Semin. Mat., Univ. Politec. Torino, Volume 44 (1986) no. 2, pp. 277-286 | MR 906018 | Zbl 0619.13005

[14] Hibi, Takayuki Distributive Lattices, Affine Semigroup Rings and Algebras with Straightening Laws, Commutative Algebra and Combinatorics (Adv. Stud. Pure Math.) Volume 11 (1987), pp. 93-109 | Article | MR 951198 | Zbl 0654.13015

[15] Kapovich, Michael; Millson, John J. The symplectic geometry of polygons in Euclidean space, J. Differ. Geom., Volume 44 (1996) no. 3, pp. 479-513 | Article | MR 1431002 | Zbl 0889.58017

[16] Kaveh, Kiumars; Manon, Christopher Khovanskii bases, higher rank valuations and tropical geometry (2016) (arXiv preprint https://arxiv.org/abs/1610.00298) | Zbl 07110730

[17] Kogan, Mikhail; Miller, Ezra Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes, Adv. Math., Volume 193 (2005) no. 1, pp. 1-17 | Article | MR 2132758 | Zbl 1084.14049

[18] Lasoń, Michał; Michałek, Mateusz On the toric ideal of a matroid, Adv. Math., Volume 259 (2014), pp. 1-12 | Article | MR 3197649 | Zbl 1297.14055

[19] Maclagan, Diane; Sturmfels, Bernd Introduction to tropical geometry, Grad. Stud. Math., Volume 161, American Mathematical Society, Providence, RI, 2015, xii+363 pages | MR 3287221 | Zbl 1321.14048

[20] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Grad. Texts Math., Volume 227, Springer-Verlag, New York, 2005, xiv+417 pages | MR 2110098 | Zbl 1090.13001

[21] Nishinou, Takeo; Nohara, Yuichi; Ueda, Kazushi Toric degenerations of Gelfand–Cetlin systems and potential functions, Adv. Math., Volume 224 (2010) no. 2, pp. 648-706 | Article | MR 2609019 | Zbl 1221.53122

[22] Ohsugi, Hidefumi; Hibi, Takayuki Toric ideals generated by quadratic binomials, J. Algebra, Volume 218 (1999) no. 2, pp. 509-527 | Article | MR 1705794 | Zbl 0943.13014

[23] Rietsch, Konstanze; Williams, Lauren Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians (2017) (arXiv preprint https://arxiv.org/abs/1712.00447)

[24] Robbiano, Lorenzo; Sweedler, Moss Subalgebra bases, Commutative Algebra (Lect. Notes Math.) Volume 1430, Springer, Berlin, Heidelberg, 1990, pp. 61-87 | Article | MR 1068324 | Zbl 0725.13013

[25] Speyer, David; Sturmfels, Bernd The tropical Grassmannian, Adv. Geom., Volume 4 (2004) no. 3, pp. 389-411 | MR 2071813 | Zbl 1065.14071

[26] Sturmfels, Bernd Gröbner bases and convex polytopes, Univ. Lect. Ser., Volume 8, American Mathematical Society, 1996 | Zbl 0856.13020

[27] Sturmfels, Bernd; Zelevinsky, Andrei Maximal minors and their leading terms, Adv. Math., Volume 98 (1993) no. 1, pp. 65-112 | Article | MR 1212627 | Zbl 0776.13009

[28] White, Neil L. A unique exchange property for bases, Linear Algebra Appl., Volume 31 (1980), pp. 81-91 | Article | MR 570381 | Zbl 0458.05022

[29] Witaszek, Jakub The degeneration of the Grassmannian into a toric variety and the calculation of the eigenspaces of a torus action, J. Algebr. Stat., Volume 6 (2015) no. 1, pp. 62-79 | MR 3361392 | Zbl 1346.14119

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