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Transformation Groups

, Volume 22, Issue 2, pp 321–352 | Cite as

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

  • EVGENY FEIGIN
  • GHISLAIN FOURIER
  • PETER LITTELMANN
Article

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.

Keywords

Toric Variety Schubert Variety Dominant Weight Fundamental Weight Dyck Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • EVGENY FEIGIN
    • 1
    • 2
  • GHISLAIN FOURIER
    • 3
    • 4
  • PETER LITTELMANN
    • 3
  1. 1.Department of MathematicsNational Research University, Higher School of EconomicsMoscowRussia
  2. 2.Tamm Theory DivisionLebedev Physics Institute of the Russian Academy of SciencesMoscowRussia
  3. 3.Mathematisches InstitutUniversität zu KölnKölnGermany
  4. 4.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

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