Hostname: page-component-7857688df4-cdr9j Total loading time: 0 Render date: 2025-11-14T19:13:01.140Z Has data issue: false hasContentIssue false
  • English
  • Français

Initial degenerations of flag varieties

Part of: Cycles and subschemes Polytopes and polyhedra Special varieties

Published online by Cambridge University Press:  10 November 2025

DANIEL COREY  and
JORGE ALBERTO OLARTE
DANIEL COREY
Affiliation:
Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, Florida, U.S.A. e-mail: daniel.corey@erau.edu
JORGE ALBERTO OLARTE
Affiliation:
Factored Cl. 7 Sur #42-145 Medellín, 050022, Colombia. e-mail: jorge.olarte@factored.ai
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\mathrm{F}\ell^{\circ}(n)$–the open subvariety of the complete flag variety $\mathrm{F}\ell(n)$ consisting of flags in general position—are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\mathrm{F}\ell(n)$ by the diagonal torus of $\textrm{PGL}(n)$ and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

MSC classification

Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 14C05: Parametrization (Chow and Hilbert schemes) 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 29
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aguiar, M. and Ardila, F.. Hopf monoids and generalised permutahedra. Mem. Amer. Math. Soc. 289 (1437) (2023), vi +119.Google Scholar
Alexeev, V. and Brion, M.. Toric degenerations of spherical varieties. Selecta Math. (N.S.). 10 (4) (2004), 453–478.10.1007/s00029-005-0396-8CrossRefGoogle Scholar
Ardila, F. and Klivans, C. J.. The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96 (1) (2006), 38–49.10.1016/j.jctb.2005.06.004CrossRefGoogle Scholar
Billera, L. J. and Sturmfels, B.. Fiber polytopes. Ann. of Math. (2). 135 (3) (1992), 527–549.10.2307/2946575CrossRefGoogle Scholar
Bossinger, L., Fras-Medina, B., Magee, T. and Nájera Chávez, A.. Toric degenerations of cluster varieties and cluster duality. Compositio Math. 156 (10) (2020), 2149–2206.10.1112/S0010437X2000740XCrossRefGoogle Scholar
Bossinger, L., Lamboglia, S., Mincheva, K. and Mohammadi, F.. Computing toric degenerations of flag varieties. In Combinatorial Algebraic Geometry. Fields Inst. Commun. vol 80 (Fields Inst. Res. Math. Sci., Toronto, ON, 2017), 247–281.10.1007/978-1-4939-7486-3_12CrossRefGoogle Scholar
Brandt, M., Eur, C. and Zhang, L.. Tropical flag varieties. Adv. Math. 384 (2021), 107695.10.1016/j.aim.2021.107695CrossRefGoogle Scholar
Caldero, P.. Toric degenerations of Schubert varieties. Transform. Groups. 7 (1) (2002), 51–60.10.1007/BF01253464CrossRefGoogle Scholar
Corey, D.. Initial degenerations of Grassmannians. Sel. Math. New Ser. 27 (57) (2021).10.1007/s00029-021-00679-6CrossRefGoogle Scholar
Corey, D.. Initial degenerations of spinor varieties. Eur. J. Math. 10(1) (2024), Paper No. 4, 41.10.1007/s40879-023-00715-zCrossRefGoogle Scholar
Corey, D. and Luber, D.. The Grassmannian of 3-planes in $\Bbb C^8$ is schön. Algebr. Comb. 6 (5) (2023), 1273–1299.Google Scholar
De Loera, J. A., Rambau, J. and Santos, F.. Triangulations. Algorithms Comput. Math. vol 25 (Springer-Verlag, Berlin, 2010). Structures for algorithms and applications.10.1007/978-3-642-12971-1CrossRefGoogle Scholar
Dress, A. W. M. and Wenzel, W.. Valuated matroids. Adv. Math. 93 (2) (1992), 214–250.10.1016/0001-8708(92)90028-JCrossRefGoogle Scholar
Eder, C., Decker, W., Fieker, C., Horn, M. and Joswig, M., editors. The OSCAR Book (Springer, Cham, 2024).Google Scholar
Fang, X., Fourier, G. and Littelmann, P.. On toric degenerations of flag varieties. In Representation Theory—Current trends and Perspectives. EMS Ser. Congr. Rep. (Eur. Math. Soc., Zürich, 2017), pp. 187–232.10.4171/171-1/8CrossRefGoogle Scholar
Fulton, W.. Young Tableaux. London Math. Soc. Stud. Texts vol. 35 (Cambridge University Press, Cambridge, 1997). With applications to representation theory and geometry.10.1017/CBO9780511626241CrossRefGoogle Scholar
Gawrilow, E. and Joswig, M.. Polymake: a Framework for Analyzing Convex Polytopes. (Birkhäuser Basel, Basel, 2000), 43–73.Google Scholar
Gross, M., Hacking, P., Keel, S. and Kontsevich, M.. Canonical bases for cluster algebras. J. Amer. Math. Soc. 31 (2) (2018), 497–608.10.1090/jams/890CrossRefGoogle Scholar
Hacking, P., Keel, S. and Tevelev, J.. Stable pair, tropical and log canonical compactifications of moduli spaces of del Pezzo surfaces. Invent. Math. 178 (1) (2009), 173–227.10.1007/s00222-009-0199-1CrossRefGoogle Scholar
Harada, M. and Kaveh, K.. Integrable systems, toric degenerations and Okounkov bodies. Invent. Math. 202 (3) (2015), 927–985.10.1007/s00222-014-0574-4CrossRefGoogle Scholar
Helm, D. and Katz, E.. Monodromy filtrations and the topology of tropical varieties. Canad. J. Math. 64 (4) (2012), 845–868.10.4153/CJM-2011-067-9CrossRefGoogle Scholar
Herrmann, S., Jensen, A., Joswig, M. and Sturmfels, B.. How to draw tropical planes. The Electron. J. Combin. 16 (2) (2009).10.37236/72CrossRefGoogle Scholar
Hu, Y.. The geometry and topology of quotient varieties of torus actions. Duke Math. J. 68 (1) (1992), 151–184.Google Scholar
Huber, B., Rambau, J. and Santos, F.. The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS). 2 (2) (2000), 179–198.10.1007/s100970050003CrossRefGoogle Scholar
Joswig, M., Loho, G., Luber, D. and Olarte, J. A.. Generalised permutahedra and positive flag dressians. Int. Math. Res. Not. IMRN. (19) (2023), 16748–16777.10.1093/imrn/rnac349CrossRefGoogle Scholar
Kaluba, M., B. Lorenz and S. Timme. Polymake.jl: A new interface to polymake. In A. M. Bigatti, J. Carette, J. H. Davenport, M. Joswig and T. de Wolff, editors, Mathematical Software – ICMS 2020. (Springer International Publishing, Cham, 2020), 377–385.Google Scholar
Kapranov, M. M., Sturmfels, B. and Zelevinsky, A. V.. Quotients of toric varieties. Math. Ann. 290 (4) (1991), 643–655.10.1007/BF01459264CrossRefGoogle Scholar
Kaveh, K.. Crystal bases and Newton–Okounkov bodies. Duke Math. J. 164 (13) (2015), 2461–2506.10.1215/00127094-3146389CrossRefGoogle Scholar
Keel, S. and Tevelev, J.. Geometry of Chow quotients of Grassmannians. Duke Math. J. 134 (2) (2006), 259–311.10.1215/S0012-7094-06-13422-1CrossRefGoogle Scholar
Kogan, M. and Miller, E.. Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes. Adv. Math. 193 (1) (2005), 1–17.10.1016/j.aim.2004.03.017CrossRefGoogle Scholar
Luxton, M. and Qu, Z.. Some results on tropical compactifications. Trans. Amer. Math. Soc. 363 (9) (2011), 4853–4876.10.1090/S0002-9947-2011-05254-2CrossRefGoogle Scholar
Maclagan, D. and Sturmfels, B.. Introduction to Tropical Geometry. Grad. Stud. Math. vol 161. (American Mathematical Society, Providence, RI, 2015).Google Scholar
Oscar – open source computer algebra research system, version 0.8.3-dev, 2022.Google Scholar
Oxley, J.. Matroid Theory. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1992).Google Scholar
Rietsch, K. and Williams, L.. Newton–Okounkov bodies, cluster duality and mirror symmetry for Grassmannians. Duke Math. J. 168 (18) (2019), 3437–3527.10.1215/00127094-2019-0028CrossRefGoogle Scholar
Schock, N.. Quasilinear tropical compactifications Preprint: arXiv:2112.02062 (2021).Google Scholar
Speyer, D.. Tropical linear spaces. SIAM J. Discrete Math. 22 (4) (2008), 1527–1558.10.1137/080716219CrossRefGoogle Scholar
Sturmfels, B.. Solving systems of polynomial equations. CBMS Reg. Conf. Ser. Math. vol. 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 2002).10.1090/cbms/097CrossRefGoogle Scholar
Tevelev, J.. Compactifications of subvarieties of tori. Amer. J. Math. 129 (4) (2007), 1087–1104.10.1353/ajm.2007.0029CrossRefGoogle Scholar
White, N.. Theory of Matroids. Encyclopedia of Math. Appl. vol. 26 (Cambridge University Press, 1986).Google Scholar