Bossinger, L.; Fang, Xin; Fourier, Ghislain; Hering, Milena; Lanini, Martina Toric degenerations of \(\mathrm{Gr}(2, n)\) and \(\mathrm{Gr}(3, 6)\) via plabic graphs. (English) Zbl 06948249 Ann. Comb. 22, No. 3, 491-512 (2018). Summary: We establish an explicit bijection between the toric degenerations of the Grassmannian \(\mathrm{Gr}(2, n)\) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to \(\mathrm{Gr}(2, n)\). We show that a similar statement does not hold for \(\mathrm{Gr}(3, 6)\). Cited in 4 Documents MSC: 14M15 Grassmannians, Schubert varieties, flag manifolds 14T05 Tropical geometry (MSC2010) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 05C21 Flows in graphs Keywords:Grassmannians; toric varieties; tropical varieties; Groebner degenerations; plabic graphs Software:Macaulay2 PDF BibTeX XML Cite \textit{L. Bossinger} et al., Ann. Comb. 22, No. 3, 491--512 (2018; Zbl 06948249) Full Text: DOI References: [1] Anderson, D., Okounkov bodies and toric degenerations, Math. Ann., 356, 1183-1202, (2013) · Zbl 1273.14104 [2] Brodsky, S.; Ceballos, C.; Labbé, J., Cluster algebras of type D4, tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom., 58, 25-46, (2017) · Zbl 1401.13063 [3] Buczyńska, W.; Wiśniewski, J., On geometry of binary symmetric models of phylogenetic trees, J. Eur. Math. Soc., 9, 609-635, (2007) · Zbl 1147.14027 [4] Eisenbud, D.: Commutative Algebra. Grad. Texts in Math., Vol. 150. Springer-Verlag, New York (1995) · Zbl 0819.13001 [5] Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/ [6] Gross, M.; Hacking, P.; Keel, S., Birational geometry of cluster algebras, Algebr. Geom., 2, 137-175, (2015) · Zbl 1322.14032 [7] Gross, M.; Hacking, P.; Keel, S.; Kontsevich, M., Canonical bases for cluster algebras, J. Amer. Math. Soc., 31, 497-608, (2018) · Zbl 1446.13015 [8] Hering, M.: Macaulay 2 computation comparing toric degenerations of Gr(3, 6) arising from plabic graphs with the tropical Grassmannian trop Gr(3, 6). Available at http://www.maths.ed.ac.uk/ mhering/Papers/PlabicGraphDegens/PlabicGraphDegensGr36.m2 [9] Kaveh, K.; Khovanskii, A. G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176, 925-978, (2012) · Zbl 1270.14022 [10] Kaveh, K., Manon, C.: Khovanskii bases, higher rank valuations and tropical geometry. Preprint (2016) [11] Kodama, Y.; Williams, L., KP solitons and total positivity for the Grassmannian, Invent. Math., 198, 637-699, (2014) · Zbl 1306.35109 [12] Lakshmibai, V., Brown, J.: The Grassmannian Variety. Geometric and Representation-Theoretic Aspects. Dev. Math., Vol. 42. Springer, New York (2015) · Zbl 1343.14001 [13] Lazarsfeld, R.; Mustaţă, M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42, 783-835, (2009) · Zbl 1182.14004 [14] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Grad. Stud. Math., Vol. 161. American Mathematical Society, Providence, RI (2015) · Zbl 1321.14048 [15] Postnikov, A.: Total positivity, Grassmannians, and networks. Preprint (2006) [16] Rietsch, K.,Williams, L.: Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. Preprint (2017) [17] Scott, J. S., Grassmannians and cluster algebras, Proc. London Math. Soc. (3), 92, 345-380, (2006) · Zbl 1088.22009 [18] Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 389-411, (2004) · Zbl 1065.14071 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.