Abstract
For classical groups \(SL_n(\mathbb {C})\), \(SO_n(\mathbb {C})\) and \(Sp_{2n}(\mathbb {C})\), we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand–Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton–Okounkov polytopes with the Feigin–Fourier–Littelmann–Vinberg polytopes. In types B and D, we compute low-dimensional examples and formulate open questions.
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Notes
Das Problem besteht darin, diejenigen geometrischen Anzahlen strenge und unter genauer Feststellung der Grenzen ihrer Gültigkeit zu beweisen, die insbesondere Schubert auf Grund des sogenannten Princips der speciellen Lage mittelst des von ihm ausgebildeten Abzählungskalküls bestimmt hat (Hilbert).
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I am grateful to the referee for the careful reading of the paper and useful comments.
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To my teacher R. K. Gordin with gratitude and admiration.
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I was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project “5-100” and by RSF grant 19-11-00056.
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Kiritchenko, V. Newton–Okounkov Polytopes of Flag Varieties for Classical Groups. Arnold Math J. 5, 355–371 (2019). https://doi.org/10.1007/s40598-019-00125-8
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DOI: https://doi.org/10.1007/s40598-019-00125-8