Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes. (English) Zbl 1522.14065
The author investigates Newton-Okounkov bodies of the flag variety \(G/B\) for \(G = \mathrm{SL}_{n+1}(\mathbb{C})\) and \(B\) the subgroup of upper triangular matrices. It is already known that several classes of polytopes can be realised as Newton-Okounkov bodies of flag varieties. These are Gelfand-Tsetlin polytopes, Berenstein-Littlemann-Zelevinsky’s string polytopes, Nakashima-Zelevinsky polytopes, Lusztig polytopes, Feigin-Fourier-Littelmann-Vinberg polytopes and polytopes constructed from extended \(g\)-vectors in cluster theory. The main result of the paper is that the class of marked chain-order polytopes, constructed from the Gelfand-Tsetlin posets of type A, can be added to this list: they can be realised as Newton-Okounkov bodies \(\Delta(G/B, \mathcal{L}_{\lambda}, \nu)\) for certain line bundle \(\mathcal{L}_{\lambda}\) and valuation \(\nu\). Moreover, the author uses this result to prove that for a marked chain-order polytope \(\Delta_{\mathcal{C}, \mathcal{O}}\) there is a flat degeneration of \(G/B\) to a normal projective toric variety corresponding to \(\Delta_{\mathcal{C}, \mathcal{O}}\).
Reviewer: Maria Donten-Bury (Warszawa)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 05E10 | Combinatorial aspects of representation theory |
| 06A07 | Combinatorics of partially ordered sets |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
Keywords:
Newton-Okounkov body; marked chain-order polytope; flag variety; toric degeneration; essential basisReferences:
| [1] | Anderson, Dave, Okounkov bodies and toric degenerations, Math. Ann., 1183-1202 (2013) · Zbl 1273.14104 · doi:10.1007/s00208-012-0880-3 |
| [2] | Ardila, Federico, Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as marked poset polytopes, J. Combin. Theory Ser. A, 2454-2462 (2011) · Zbl 1234.52009 · doi:10.1016/j.jcta.2011.06.004 |
| [3] | Brion, Michel, Topics in cohomological studies of algebraic varieties. Lectures on the geometry of flag varieties, Trends Math., 33-85 (2005), Birkh\"{a}user, Basel · Zbl 1487.14105 · doi:10.1007/3-7643-7342-3\_2 |
| [4] | Cox, David A., Toric varieties, Graduate Studies in Mathematics, xxiv+841 pp. (2011), American Mathematical Society, Providence, RI · Zbl 1223.14001 · doi:10.1090/gsm/124 |
| [5] | Fang, Xin, Marked chain-order polytopes, European J. Combin., 267-282 (2016) · Zbl 1343.05163 · doi:10.1016/j.ejc.2016.06.007 |
| [6] | Fang, Xin, Essential bases and toric degenerations arising from birational sequences, Adv. Math., 107-149 (2017) · Zbl 1453.17005 · doi:10.1016/j.aim.2017.03.014 |
| [7] | Fang, Xin, A continuous family of marked poset polytopes, SIAM J. Discrete Math., 611-639 (2020) · Zbl 1445.52009 · doi:10.1137/18M1228529 |
| [8] | Fang, Xin, The Minkowski property and reflexivity of marked poset polytopes, Electron. J. Combin., Paper No. 1.27, 19 pp. (2020) · Zbl 1437.52011 |
| [9] | Feigin, Evgeny, PBW filtration and bases for irreducible modules in type \({\mathsf A}_n\), Transform. Groups, 71-89 (2011) · Zbl 1237.17011 · doi:10.1007/s00031-010-9115-4 |
| [10] | Feigin, Evgeny, PBW filtration and bases for symplectic Lie algebras, Int. Math. Res. Not. IMRN, 5760-5784 (2011) · Zbl 1233.17007 · doi:10.1093/imrn/rnr014 |
| [11] | Feigin, Evgeny, Favourable modules: filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, Transform. Groups, 321-352 (2017) · Zbl 1461.14068 · doi:10.1007/s00031-016-9389-2 |
| [12] | Fourier, Ghislain, Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence, J. Pure Appl. Algebra, 606-620 (2016) · Zbl 1328.52007 · doi:10.1016/j.jpaa.2015.07.007 |
| [13] | Fujita, Naoki, Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules, J. Algebra, 408-447 (2018) · Zbl 1467.17009 · doi:10.1016/j.jalgebra.2018.08.019 |
| [14] | Fujita, Naoki, Polyhedral realizations of crystal bases and convex-geometric Demazure operators, Selecta Math. (N.S.), Paper No. 74, 35 pp. (2019) · Zbl 1428.05315 · doi:10.1007/s00029-019-0522-7 |
| [15] | Fujita, Naoki, Newton-Okounkov bodies of flag varieties and combinatorial mutations, Int. Math. Res. Not. IMRN, 9567-9607 (2021) · Zbl 1524.14109 · doi:10.1093/imrn/rnaa276 |
| [16] | Fujita, Naoki, Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases, Math. Z., 325-352 (2017) · Zbl 1395.17026 · doi:10.1007/s00209-016-1709-7 |
| [17] | Fujita, Naoki, A comparison of Newton-Okounkov polytopes of Schubert varieties, J. Lond. Math. Soc. (2), 201-227 (2017) · Zbl 1427.17024 · doi:10.1112/jlms.12059 |
| [18] | Naoki Fujita and Hironori Oya, Newton-Okounkov polytopes of Schubert varieties arising from cluster structures, Preprint, 2002.09912v1, 2020. · Zbl 1427.17024 |
| [19] | Gel\cprime fand, I. M., Finite-dimensional representations of the group of unimodular matrices, Doklady Akad. Nauk SSSR (N.S.), 825-828 (1950) · Zbl 0037.15301 |
| [20] | Harada, Megumi, Integrable systems, toric degenerations and Okounkov bodies, Invent. Math., 927-985 (2015) · Zbl 1348.14122 · doi:10.1007/s00222-014-0574-4 |
| [21] | Jantzen, Jens Carsten, Representations of algebraic groups, Mathematical Surveys and Monographs, xiv+576 pp. (2003), American Mathematical Society, Providence, RI · Zbl 1034.20041 |
| [22] | Kashiwara, Masaki, Representations of groups. On crystal bases, CMS Conf. Proc., 155-197 (1994), Amer. Math. Soc., Providence, RI · Zbl 0851.17014 |
| [23] | Kashiwara, Masaki, Crystal graphs for representations of the \(q\)-analogue of classical Lie algebras, J. Algebra, 295-345 (1994) · Zbl 0808.17005 · doi:10.1006/jabr.1994.1114 |
| [24] | Kaveh, Kiumars, Crystal bases and Newton-Okounkov bodies, Duke Math. J., 2461-2506 (2015) · Zbl 1428.14083 · doi:10.1215/00127094-3146389 |
| [25] | Kaveh, Kiumars, Perspectives in analysis, geometry, and topology. Algebraic equations and convex bodies, Progr. Math., 263-282 (2012), Birkh\"{a}user/Springer, New York · Zbl 1316.52011 · doi:10.1007/978-0-8176-8277-4\_12 |
| [26] | Kaveh, Kiumars, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 925-978 (2012) · Zbl 1270.14022 · doi:10.4007/annals.2012.176.2.5 |
| [27] | Kiritchenko, Valentina, Newton-Okounkov polytopes of flag varieties, Transform. Groups, 387-402 (2017) · Zbl 1396.14047 · doi:10.1007/s00031-016-9372-y |
| [28] | Kiritchenko, Valentina, Newton-Okounkov polytopes of flag varieties for classical groups, Arnold Math. J., 355-371 (2019) · Zbl 1436.52015 · doi:10.1007/s40598-019-00125-8 |
| [29] | Kumar, Shrawan, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, xvi+606 pp. (2002), Birkh\"{a}user Boston, Inc., Boston, MA · Zbl 1026.17030 · doi:10.1007/978-1-4612-0105-2 |
| [30] | Lakshmibai, Venkatramani, Standard monomial theory, Encyclopaedia of Mathematical Sciences, xiv+265 pp. (2008), Springer-Verlag, Berlin · Zbl 1137.14036 |
| [31] | Lazarsfeld, Robert, Convex bodies associated to linear series, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4), 783-835 (2009) · Zbl 1182.14004 · doi:10.24033/asens.2109 |
| [32] | Littelmann, Peter, Cones, crystals, and patterns, Transform. Groups, 145-179 (1998) · Zbl 0908.17010 · doi:10.1007/BF01236431 |
| [33] | Okounkov, Andrei, Brunn-Minkowski inequality for multiplicities, Invent. Math., 405-411 (1996) · Zbl 0893.52004 · doi:10.1007/s002220050081 |
| [34] | Okounkov, Andrei, Kirillov’s seminar on representation theory. Multiplicities and Newton polytopes, Amer. Math. Soc. Transl. Ser. 2, 231-244 (1998), Amer. Math. Soc., Providence, RI · Zbl 0920.20032 · doi:10.1090/trans2/181/07 |
| [35] | Okounkov, Andrei, The orbit method in geometry and physics. Why would multiplicities be log-concave?, Progr. Math., 329-347 (2000), Birkh\"{a}user Boston, Boston, MA · Zbl 1063.22024 |
| [36] | Stanley, Richard P., Two poset polytopes, Discrete Comput. Geom., 9-23 (1986) · Zbl 0595.52008 · doi:10.1007/BF02187680 |
| [37] | Ernest B. Vinberg, On some canonical bases of representation spaces of simple Lie algebras, Conference Talk, Bielefeld, 2005. |
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