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Newton-Okounkov polytopes of flag varieties. (English) Zbl 1396.14047
The author computes for a valuation coming from a certain flag of (Schubert-)subvarieties for the full flag variety of type A the corresponding Newton-Okounkov bodies. The main result of the paper is that these Newton-Okounkov bodies coincide with the Feigin-Fourier-Littelmann-Vinberg polytopes.
Newton-Okounkov bodies are a far generalization of Newton polytopes and were introduced to study the asymptotic behavior of line bundles (as introduced by independently Okounkov, Lazarsfeld-Mustata, Kaveh-Khovanskii). For the full flag variety of type A Kiritchenko chooses the reduced expression \((s_1)(s_2s_1)\dots(s_{n-1}s_1)\) of the longest element in the symmetric group Sn. The reduced expression determines a flag of translated Schubert varieties in \(\mathrm{GL}_n/B\) (see (*) in the paper). Following constructions of Anderson and Kaveh, such a flag of subvarieties determines a valuation on the field of rational functions on \(\mathrm{GL}_n/B\), which can alternatively be defined by a lowest term order (see Section 2.1). With respect to these valuations Kiritchenko computes Newton-Okounkov bodies for line bundles associated to dominant weights.
To compute explicit examples of the Newton-Okounkov bodies, in Section 2.2 the author introduces coordinates that are compatible with the flag (*). These are closely related to the work of Vakil establishing a geometric Littlewood Richardson rule for Grassmannians. Among the explicit examples presented in Section 2.3 are \(\mathrm{GL}_4/B\) and Grassmannians.
In the following Section 3 the proof of the main result is given. Using general properties of Newton-Okounkov bodies and their compatibility with subvarieties (Lemma 3.2) Kiritchenko takes slices (flags of faces) of both, the Newton-Okounkov body and the FFLV polytopes. The faces are described inductively using convex-geometric methods for the case of FFLV polytopes (Lemma 3.6) and algebro-geometric methods for the Newton Okounkov bodies (Lemma 3.7). The aforementioned Lemmata are used to perform induction in the proof of Theorem 3.5, stating that the corresponding slices of the FFLV polytope and the Newton-Okounkov body coincide. Applied to the trivial slice (the whole polytope) this specializes to the main result. This result is only true in type A. In type C, for which FFLV polytopes are also defined it fails as Kiritchenko shows in Section 2.4 analyzing the case of \(\mathrm{Sp}_4\) in detail.
The last section of the paper studies combinatorial properties of the FFLV polytope and relates to the Gelfand-Tsetlin polytope. Considering flags of faces of the Gelfand-Tsetlin polytope, Kiritchenko proves that the corresponding faces of both polytopes have the same Ehrhardt polynomial. Combining with a previous result of the author this implies in particular that the face polytopes of the FFLV flags have as many integer points as the dimension of certain Demazure modules.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
20G05 Representation theory for linear algebraic groups
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
Full Text: DOI
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