String cone and superpotential combinatorics for flag and Schubert varieties in type A. (English) Zbl 1417.05245
Summary: We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by M. Gross et al. [J. Am. Math. Soc. 31, No. 2, 497–608 (2018; Zbl 1446.13015)]: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary.
We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero’s toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory.
We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero’s toric degenerations of Schubert varieties as GHKK-degeneration using cluster theory.
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 13F60 | Cluster algebras |
Keywords:
Schubert variety; flag variety; superpotential; string cone; mirror symmetry; pseudoline arrangement; combinatoricsCitations:
Zbl 1446.13015References:
| [1] | Anderson, Dave, Okounkov bodies and toric degenerations, Math. Ann., 356, 3, 1183-1202 (2013) · Zbl 1273.14104 |
| [2] | Berenstein, Arkady; Zelevinsky, Andrei, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., 143, 1, 77-128 (2001) · Zbl 1061.17006 |
| [3] | Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei, Parametrizations of canonical bases and totally positive matrices, Adv. Math., 122, 1, 49-149 (1996) · Zbl 0966.17011 |
| [4] | Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., 126, 1, 1-52 (2005) · Zbl 1135.16013 |
| [5] | Bossinger, Lara; Frías-Medina, Bosco; Magee, Timothy; Nájera Chávez, Alfredo, Toric degenerations of cluster varieties and cluster duality (2018), arXiv preprint · Zbl 1454.13030 |
| [8] | Fang, Xin; Fourier, Ghislain; Littelmann, Peter, Essential bases and toric degenerations arising from birational sequences, Adv. Math., 312, 107-149 (2017) · Zbl 1453.17005 |
| [9] | Fock, V. V.; Goncharov, A. B., Cluster \(X\)-varieties, amalgamation, and Poisson-Lie groups, (Algebraic Geometry and Number Theory. Algebraic Geometry and Number Theory, Progr. Math., vol. 253 (2006), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 27-68 · Zbl 1162.22014 |
| [10] | Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529 (2002) · Zbl 1021.16017 |
| [11] | Fomin, Sergey; Williams, Lauren; Zelevinsky, Andrei, Introduction to cluster algebras. chapters 1-3 (2016), arXiv preprint |
| [12] | Gawrilow, Ewgenij; Joswig, Michael, Polymake: a framework for analyzing convex polytopes, (Polytopes-Combinatorics and Computation (2000), Springer), 43-73 · Zbl 0960.68182 |
| [13] | Geiss, Christof; Leclerc, Bernard; Schröer, Jan, Cluster algebras in algebraic Lie theory, Transform. Groups, 18, 1, 149-178 (2013) · Zbl 1278.13027 |
| [14] | Gelfand, I. M.; Tsetlin, M. L., Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR, 71, 825-828 (1950) · Zbl 0037.15301 |
| [16] | Genz, Volker; Koshevoy, Gleb; Schumann, Bea, Polyhedral parametrizations of canonical bases & cluster duality (2017), arXiv preprint · Zbl 1453.13065 |
| [17] | Gleizer, Oleg; Postnikov, Alexander, Littlewood-Richardson coefficients via Yang-Baxter equation, Int. Math. Res. Not., 2000, 14, 741-774 (2000) · Zbl 0987.20023 |
| [18] | Goncharov, Alexander; Shen, Linhui, Geometry of canonical bases and mirror symmetry, Invent. Math., 202, 2, 487-633 (2015) · Zbl 1355.14030 |
| [19] | Gonciulea, Nicolae; Lakshmibai, Venkatramani, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups, 1, 3, 215-248 (1996) · Zbl 0909.14028 |
| [20] | Gross, Mark; Hacking, Paul; Keel, Sean, Birational geometry of cluster algebras, Algebr. Geom., 2, 2, 137-175 (2015) · Zbl 1322.14032 |
| [21] | Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim, Canonical bases for cluster algebras, J. Amer. Math. Soc., 31, 2, 497-608 (2018) · Zbl 1446.13015 |
| [22] | Hibi, Takayuki; Li, Nan, Unimodular equivalence of order and chain polytopes, Math. Scand., 118, 1, 5-12 (2016) · Zbl 1335.52026 |
| [23] | Kaveh, Kiumars, Crystal bases and Newton-Okounkov bodies, Duke Math. J., 164, 13, 2461-2506 (2015) · Zbl 1428.14083 |
| [25] | Kaveh, Kiumars; Manon, Christopher, Khovanskii bases, higher rank valuations and tropical geometry (2016), arXiv preprint · Zbl 1420.14146 |
| [26] | Keller, Bernhard, Quiver mutation in JavaScript and Java, Available at |
| [28] | Littelmann, Peter, Cones, crystals, and patterns, Transform. Groups, 3, 2, 145-179 (1998) · Zbl 0908.17010 |
| [29] | Maclagan, Diane; Sturmfels, Bernd, Introduction to Tropical Geometry, Graduate Studies in Mathematics, vol. 161 (2015), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1321.14048 |
| [32] | Scott, Joshua S., Grassmannians and cluster algebras, Proc. Lond. Math. Soc. (3), 92, 2, 345-380 (2006) · Zbl 1088.22009 |
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