## Reflexivity of Newton–Okounkov bodies of partial flag varieties

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- by Christian Steinert
- Represent. Theory
**26**(2022), 859-873 - DOI: https://doi.org/10.1090/ert/621
- Published electronically: August 16, 2022
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## Abstract:

Assume that the valuation semigroup $\Gamma (\lambda )$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal {L_\lambda }$ constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if $\mathcal {L}_\lambda$ is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive.## References

- Shreeram Abhyankar,
*On the valuations centered in a local domain*, Amer. J. Math.**78**(1956), 321–348. MR**82477**, DOI 10.2307/2372519 - Valery Alexeev and Michel Brion,
*Toric degenerations of spherical varieties*, Selecta Math. (N.S.)**10**(2004), no. 4, 453–478. MR**2134452**, DOI 10.1007/s00029-005-0396-8 - Dave Anderson,
*Okounkov bodies and toric degenerations*, Math. Ann.**356**(2013), no. 3, 1183–1202. MR**3063911**, DOI 10.1007/s00208-012-0880-3 - Victor V. Batyrev,
*Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*, J. Algebraic Geom.**3**(1994), no. 3, 493–535. MR**1269718** - Victor V. Batyrev,
*Toric degenerations of Fano varieties and constructing mirror manifolds*, Univ. Torino, Turin, 2004. - Victor V. Batyrev, Ionuţ Ciocan-Fontanine, Bumsig Kim, and Duco van Straten,
*Mirror symmetry and toric degenerations of partial flag manifolds*, Acta Math.**184**(2000), no. 1, 1–39. MR**1756568**, DOI 10.1007/BF02392780 - Matthias Beck and Sinai Robins,
*Computing the continuous discretely*, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. MR**2271992** - A. D. Berenstein and A. V. Zelevinsky,
*Tensor product multiplicities and convex polytopes in partition space*, J. Geom. Phys.**5**(1988), no. 3, 453–472. MR**1048510**, DOI 10.1016/0393-0440(88)90033-2 - Arkady Berenstein and Andrei Zelevinsky,
*Tensor product multiplicities, canonical bases and totally positive varieties*, Invent. Math.**143**(2001), no. 1, 77–128. MR**1802793**, DOI 10.1007/s002220000102 - Philippe Caldero,
*Toric degenerations of Schubert varieties*, Transform. Groups**7**(2002), no. 1, 51–60. MR**1888475**, DOI 10.1007/s00031-002-0003-4 - Xin Fang, Ghislain Fourier, and Peter Littelmann,
*Essential bases and toric degenerations arising from birational sequences*, Adv. Math.**312**(2017), 107–149. MR**3635807**, DOI 10.1016/j.aim.2017.03.014 - Evgeny Feigin, Ghislain Fourier, and Peter Littelmann,
*PBW filtration and bases for irreducible modules in type ${\mathsf A}_n$*, Transform. Groups**16**(2011), no. 1, 71–89. MR**2785495**, DOI 10.1007/s00031-010-9115-4 - Evgeny Feigin, Ghislain Fourier, and Peter Littelmann,
*PBW filtration and bases for symplectic Lie algebras*, Int. Math. Res. Not.**24**(2011), 5760–5784. - Evgeny Feigin, Ghislain Fourier, and Peter Littelmann,
*Favourable modules: filtrations, polytopes, Newton-Okounkov bodies and flat degenerations*, Transform. Groups**22**(2017), no. 2, 321–352. MR**3649457**, DOI 10.1007/s00031-016-9389-2 - Naoki Fujita and Satoshi Naito,
*Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases*, Math. Z.**285**(2017), no. 1-2, 325–352. MR**3598814**, DOI 10.1007/s00209-016-1709-7 - I. M. Gel′fand and M. L. Cetlin,
*Finite-dimensional representations of the group of unimodular matrices*, Doklady Akad. Nauk SSSR (N.S.)**71**(1950), 825–828 (Russian). MR**0035774** - N. Gonciulea and V. Lakshmibai,
*Degenerations of flag and Schubert varieties to toric varieties*, Transform. Groups**1**(1996), no. 3, 215–248. MR**1417711**, DOI 10.1007/BF02549207 - A. A. Gornitskiĭ,
*Essential signatures and canonical bases of irreducible representations of the group $G_2$*, Mat. Zametki**97**(2015), no. 1, 35–47 (Russian, with Russian summary); English transl., Math. Notes**97**(2015), no. 1-2, 30-41. MR**3370491**, DOI 10.4213/mzm10384 - A. Gornitskii,
*Essential signatures and canonical bases for ${B}_n$ and ${D}_n$*, arXiv:1611.07381, 2016. - Takayuki Hibi,
*Dual polytopes of rational convex polytopes*, Combinatorica**12**(1992), no. 2, 237–240. MR**1179260**, DOI 10.1007/BF01204726 - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR**0323842**, DOI 10.1007/978-1-4612-6398-2 - Kiumars Kaveh,
*Crystal bases and Newton-Okounkov bodies*, Duke Math. J.**164**(2015), no. 13, 2461–2506. MR**3405591**, DOI 10.1215/00127094-3146389 - Kiumars Kaveh and A. G. Khovanskii,
*Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory*, Ann. of Math. (2)**176**(2012), no. 2, 925–978. MR**2950767**, DOI 10.4007/annals.2012.176.2.5 - Kiumars Kaveh and Christopher Manon,
*Khovanskii bases, higher rank valuations, and tropical geometry*, SIAM J. Appl. Algebra Geom.**3**(2019), no. 2, 292–336. MR**3949692**, DOI 10.1137/17M1160148 - K. Kaveh and E. Villella,
*On a notion of anticanonical class for families of convex polytopes*, arXiv:1802.06674, 2018. - Valentina Kiritchenko,
*Newton-Okounkov polytopes of flag varieties*, Transform. Groups**22**(2017), no. 2, 387–402. MR**3649460**, DOI 10.1007/s00031-016-9372-y - Mikhail Kogan and Ezra Miller,
*Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes*, Adv. Math.**193**(2005), no. 1, 1–17. MR**2132758**, DOI 10.1016/j.aim.2004.03.017 - Bertram Kostant,
*Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. of Math. (2)**74**(1961), 329–387. MR**142696**, DOI 10.2307/1970237 - Robert Lazarsfeld and Mircea Mustaţă,
*Convex bodies associated to linear series*, Ann. Sci. Éc. Norm. Supér. (4)**42**(2009), no. 5, 783–835 (English, with English and French summaries). MR**2571958**, DOI 10.24033/asens.2109 - P. Littelmann,
*Cones, crystals, and patterns*, Transform. Groups**3**(1998), no. 2, 145–179. MR**1628449**, DOI 10.1007/BF01236431 - G. Lusztig,
*Canonical bases arising from quantized enveloping algebras*, J. Amer. Math. Soc.**3**(1990), no. 2, 447–498. MR**1035415**, DOI 10.1090/S0894-0347-1990-1035415-6 - Toshiki Nakashima and Andrei Zelevinsky,
*Polyhedral realizations of crystal bases for quantized Kac-Moody algebras*, Adv. Math.**131**(1997), no. 1, 253–278. MR**1475048**, DOI 10.1006/aima.1997.1670 - Andrei Okounkov,
*Brunn-Minkowski inequality for multiplicities*, Invent. Math.**125**(1996), no. 3, 405–411. MR**1400312**, DOI 10.1007/s002220050081 - Andreĭ Okounkov,
*Multiplicities and Newton polytopes*, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 231–244. MR**1618759**, DOI 10.1090/trans2/181/07 - K. Rietsch and L. Williams,
*Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians*, Duke Math. J.**168**(2019), no. 18, 3437–3527. MR**4034891**, DOI 10.1215/00127094-2019-0028 - J. Rusinko,
*Equivalence of mirror families constructed from toric degenerations of flag varieties*, Transform. Groups**13**(2008), no. 1, 173–194. MR**2421321**, DOI 10.1007/s00031-008-9008-y - T. A. Springer,
*Linear algebraic groups*, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1642713**, DOI 10.1007/978-0-8176-4840-4 - C. Steinert,
*Fano varieties and Fano polytopes*, Ph.D. thesis, University of Cologne, 2020, https://kups.ub.uni-koeln.de/id/eprint/16137. - Christian Steinert,
*A diagrammatic approach to string polytopes*, Algebr. Comb.**5**(2022), no. 1, 63–91. MR**4389962**, DOI 10.5802/alco.196 - The Sage Developers,
*SageMath, the Sage Mathematics Software System*, 2017, Version 8.1, https://www.sagemath.org.

## Bibliographic Information

**Christian Steinert**- Affiliation: Department of Mathematics, Chair for Algebra and Representation Theory, RWTH Aachen University, 25042 Aachen, Germany
- MR Author ID: 1492490
- ORCID: 0000-0002-5947-7298
- Email: steinert@art.rwth-aachen.de
- Received by editor(s): February 20, 2022
- Received by editor(s) in revised form: May 12, 2022
- Published electronically: August 16, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 859-873 - MSC (2020): Primary 17B10; Secondary 14L35, 14M25, 52B20
- DOI: https://doi.org/10.1090/ert/621
- MathSciNet review: 4469220