Диссертация (1136188), страница 60
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Bytaking partial degenerations of SI one can construct intermediate convex chains with the same character. However, only SI might producetrue polytopes (such as Gelfand–Zetlin polytopes for G = SLn or polytope of Example 3.4 for G = Sp4 ) that represent the Weyl characters.Indeed, such a polytope must have a face Di2 (a) (in the case of SI ) or aface Di Di (a) (in the case of the i-th degeneration of SI ).
The formeris a true segment since Di2 = Di , while the latter is necessarily a virtualtrapezoid with the same character due to cancelations (cf. Figure 3).71-06.tex : 2016/10/27 (14:27)page: 184184V. KiritchenkoReferences[Andersen] H. H. Andersen, Schubert varieties and Demazure’s character formula, Invent. Math., 79 (1985), no. 3, 611–618[Anderson] D. Anderson, Okounkov bodies and toric degenerations, Math.Ann., 356 (2013), no. 3, 1183–1202[BZ] A.Berenstein, A. Zelevinsky, Tensor product multiplicities, canonicalbases and totally positive varieties, Invent.
Math. 143 (2001), no.1, 77128[GK] M. Grossberg and Y. Karshon, Bott towers, complete integrability,and the extended character of representations, Duke Math. J., 76 (1994),no. 1, 23–58.[K] K.Kaveh, Crystal basis and Newton–Okounkov bodies, Duke Math. J.,164 (2015), no. 13, 2461–2506[KKh] K. Kaveh, A.
Khovanskii, Convex bodies associated to actions of reductive groups, Moscow Math.J., 12 (2012), 369–396[PKh] A.G. Khovanskii, A.V. Pukhlikov, Finitely additive measures of virtual polytopes, St. Petersburg Mathematical Journal, 4 (1993), no.2,337–356[PKh2] A.G.Khovanskii and A.V.Pukhlikov, The Riemann–Roch theoremfor integrals and sums of quasipolynomials on virtual polytopes, Algebrai Analiz, 4 (1992), no.4, 188–216[KST] V.
Kiritchenko, E. Smirnov, V. Timorin, Schubert calculus andGelfand–Zetlin polytopes, Russian Math. Surveys, 67 (2012), no.4, 685–719[L] P. Littelmann, Cones, crystals and patterns, Transformation Groups, 3(1998), pp. 145–179[KnM] A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2), 161 (2005), 1245–1318E-mail address: vkiritch@hse.ruLaboratory of Algebraic Geometry and Faculty of MathematicsHigher School of EconomicsVavilova St. 7, 112312 Moscow, RussiaInstitute for Information Transmission Problems, Moscow, RussiaПриложение J.Статья 10.Valentina Kiritchenko “Newton-Okounkov polytopes of flag varieties”Transformation Groups Vol. 22, No. 2 , 2017, pp.
387 – 402Разрешение на копирование: Согласно Соглашению о копирайте автор статьиможет использовать принятую в журнал рукопись со следующей ссылкой:полная журнальная версия статьи доступна на сайте издательства Шпрингерhttps://link.springer.com/article/10.1007/s00031-016-9372-yНиже копируется авторская рукопись в том виде, в каком она была принята впечать.Transformation Groupsc⃝SpringerScience+Business Media New York (??)NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIESVALENTINA KIRITCHENKO∗Laboratory of Algebraic Geometry andFaculty of MathematicsNational Research University Higher Schoolof EconomicsVavilova St. 7, 117312 Moscow, RussiaandInstitute for Information TransmissionProblems, Moscow, Russiavkiritch@hse.ruAbstract. We compute the Newton–Okounkov bodies of line bundles on the complete flag variety ofGLn for a geometric valuation coming from a flag of translated Schubert subvarieties.
The Schubertsubvarieties correspond to the terminal subwords in the decomposition (s1 )(s2 s1 )(s3 s2 s1 )(. . .)(sn−1 . . . s1 )of the longest element in the Weyl group. The resulting Newton–Okounkov bodies coincide with the Feigin–Fourier–Littelmann–Vinberg polytopes in type A.1. IntroductionNewton–Okounkov convex bodies generalize Newton polytopes from toric geometry to a moregeneral algebro-geometric as well as representation-theoretic setting. In particular, Newton–Okounkov bodies of flag varieties and of Bott–Samelson resolutions for different valuations haverecently attracted much interest due to connections with representation theory and Schubert calculus.
The Newton–Okounkov body can be assigned to a line bundle on an algebraic variety X[KaKh, LM]. In contrast with Newton polytopes, Newton–Okounkov bodies depend heavily ona choice of a valuation on the field of rational functions C(X). In the case of flag varieties, it isespecially interesting to consider various geometric valuations, namely, valuations coming from acomplete flag of subvarieties pt = Yd ⊂ . . . ⊂ Y1 ⊂ Y0 = X, where d := dim X, since the resultingNewton–Okounkov convex bodies can often be identified with polytopes that arise in representationtheory.The first explicit computation of Newton–Okounkov polytopes of flag varieties is due to Okounkov [O].
For a geometric valuation, he identified Newton–Okounkov polytopes of symplecticflag varieties with symplectic Gelfand–Zetlin polytopes. Since then several other computationswere made for different valuations [An, Fu, FFL14, HY, Ka, Ki14], see also [An15, FK, SchS] forrelated results. In the present paper, we use a natural geometric valuation introduced by Andersonin [An, Section 6.4] who computed an example for GL3 . In this example, the Newton–Okounkovpolytope was identified with the 3-dimensional Gelfand–Zetlin polytope.Let X be the complete flag variety for GLn (C). We compute Newton–Okounkov convex bodiesof semiample line bundles on X for the geometric valuation coming from the flag of translatedSchubert subvarieties−1−1w0 Xid ⊂ w0 wd−1Xwd−1 ⊂ w0 wd−2Xwd−2 ⊂ .
. . ⊂ w0 w1−1 Xw1 ⊂ X,∗The research was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences(project 14-50-00150).Received . Accepted .VALENTINA KIRITCHENKOwhere w1 , w2 ,. . . , wd−1 are terminal subwords of the decomposition(s1 )(s2 s1 )(s3 s2 s1 )(. . .)(sn−1 . . . s1 )of the longest element in Sn (see Section 2.1 for a precise definition). The valuation can bealternatively described as the lowest term valuation associated with a natural coordinate systemon the open Schubert cell in X (see Section 2.2). The computation is based on simple algebrogeometric and convex-geometric arguments.
The only representation-theoretic input is the wellknown fact that the number of integer points in the Gelfand–Zetlin polytope for a dominant weightλ is equal to the dimension of the irreducible representation of GLn with the highest weight λ.Surprisingly, the resulting polytopes for n > 3 are not, in general, combinatorially equivalentto the Gelfand–Zetlin polytopes and coincide instead with Feigin–Fourier–Littelmann–Vinbergpolytopes in type A. The complete list of cases when Feigin–Fourier–Littelmann–Vinberg polytopesin type A are combinatorially equivalent to the Gelfand–Zetlin polytopes can be found in [Fo].Though Feigin–Fourier–Littelmann–Vinberg polytopes can also be defined in type C an analogousresult for Newton–Okounkov polytopes does not hold already for Sp4 (C) (see Section 2.4 formore details).
In both types A and C, Feigin–Fourier–Littelmann–Vinberg polytopes were earlierobtained as Newton–Okounkov bodies for a completely different valuation that does not come fromany decomposition of the longest element (see [FFL14, Examples 8.1,8.2]). The fact that valuationsconsidered in [FFL14] and in the present paper yield the same Newton–Okounkov polytopes servedas the starting point for the recent preprint [FaFL15], which gives a conceptual explanation forthis coincidence (see [FaFL15, Example 17]).The paper is organized as follows. In Section 2, we define the valuation, formulate the mainresult and consider several examples. Section 3 contains the proof of the main theorem modulo the result on comparison between the Gelfand–Zetlin and Feigin–Fourier–Littelmann–Vinbergpolytopes.
The latter result is explained in Section 4 using purely convex-geometric arguments.I am grateful to Alexander Esterov, Evgeny Feigin and Evgeny Smirnov for useful discussions.I would also like to thank the referee for valuable comments.2. Main resultIn this section, we define the valuation on C(X), recall the inequalities defining Feigin–Fourier–Littelmann–Vinberg polytopes and formulate the main theorem. We also define a geometricallynatural coordinate system on the open Schubert cell and use it do the simplest examples by hand.Finally, we discuss the case of symplectic flag varieties.2.1.
ValuationFix the decomposition w0 = (s1 )(s2 s1 )(s3 s2 s1 ) . . . (sn−1 . . . s1 ) of the longestelement w0 ∈ Sn .( )Here si := (i i + 1) is the i-th elementary transposition. Denote by d := n2 the length of w0 .Fix a complete flag of subspaces F • := (F 1 ⊂ F 2 ⊂ . . . ⊂ F n−1 ⊂ Cn ) (this amounts to fixinga Borel subgroup B ⊂ GLn ). In what follows, wk for k = 1,. . . , d denotes the subword of w0obtained by deleting the first k simple reflections in w0 , and wk denotes the corresponding elementof Sn . Consider the flag of translated Schubert subvarieties:−1−1w0 Xid ⊂ w0 wd−1Xwd−1 ⊂ w0 wd−2Xwd−2 ⊂ . . . ⊂ w0 w1−1 Xw1 ⊂ GLn /B,(∗)where Schubert subvarieties are taken with respect to the flag F • , i.e., Xw = BwB/B (cf.
[An,Section 6.4] and [Ka, Remark 2.3]). Let y1 , . . . , yd be coordinates on the open Schubert cell C(with respect to F • ) that are compatible with (∗), i.e., w0 wk−1 Xwk ∩ C = {y1 = . . . = yk = 0}. Apossible choice of such coordinates is described in Section 2.2.NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIESFix the lexicographic ordering on monomials in coordinates y1 , . . . , yd , i.e., y1k1 · · · y kd ≻y l1 · · · y ld iff there exists j ≤ d such that ki = li for i < j and kj > lj . Let v denote the lowestorder term valuation on C(Xw0 ) = C(GLn /B) associated with these coordinates and ordering.Let Lλ be the line bundle on GLn /B corresponding to a dominant weight λ := (λ1 , .
. . , λn ) ∈ Znof GLn (dominant means that λ1 ≥ λ2 ≥ . . . ≥ λn ). Recall that the bundle Lλ is semiample iffλ is dominant and very ample iff λ is strictly dominant, i.e., λ1 > λ2 > . . . > λn . Denote by∆v (GLn /B, Lλ ) ⊂ Rd the Newton–Okounkov convex body corresponding to GLn /B, Lλ and v(see [KaKh, LM] for a definition of Newton–Okounkov convex bodies).Theorem 2.1. The Newton–Okounkov convex body ∆v (GLn /B, Lλ ) coincides with the Feigin–Fourier–Littelmann–Vinberg polytope F F LV (λ).We now recall the definition of F F LV (λ).
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