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no. 6
Toric degenerations of Grassmannians from matching fields
Mohammadi, Fatemeh1; Shaw, Kristin2
1 University of Bristol Bristol BS8 1TW, UK
2 University of Oslo P.O. box 1053 Blindern 0316 OSLO, Norway
Algebraic Combinatorics, Volume 2 (2019) no. 6, pp. 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 Gr(3,n).

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.77
Classification: 14M15, 14M25, 14T05
Mots-clés : toric degenerations, SAGBI and Khovanskii bases, Grassmannians, tropical geometry

Mohammadi, Fatemeh 1; Shaw, Kristin 2

1 University of Bristol Bristol BS8 1TW, UK
2 University of Oslo P.O. box 1053 Blindern 0316 OSLO, Norway
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/articles/10.5802/alco.77/

[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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

[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 | Zbl

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

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

[9] Feigin, Evgeny 𝔾 a M degeneration of flag varieties, Sel. Math., New Ser., Volume 18 (2012) no. 3, pp. 513-537 | DOI | MR | Zbl

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

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