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FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS

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Abstract

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.

In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

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Authors and Affiliations

  1. Department of Mathematics, National Research University, Higher School of Economics, Vavilova str. 7, 117312, Moscow, Russia

    EVGENY FEIGIN

  2. Tamm Theory Division, Lebedev Physics Institute of the Russian Academy of Sciences, 53 Leninskii prosp., 119991, Moscow, Russia

    EVGENY FEIGIN

  3. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931, Köln, Germany

    GHISLAIN FOURIER & PETER LITTELMANN

  4. School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, UK

    GHISLAIN FOURIER

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  1. EVGENY FEIGIN
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  2. GHISLAIN FOURIER
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  3. PETER LITTELMANN
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Correspondence to EVGENY FEIGIN.

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FEIGIN, E., FOURIER, G. & LITTELMANN, P. FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS. Transformation Groups 22, 321–352 (2017). https://doi.org/10.1007/s00031-016-9389-2

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