Favourable Modules: Filtrations, Polytopes, Newton–okounkov Bodies and Flat Degenerations

Abstract

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 flat degenerations of the corresponding flag 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.

The Minkowski Property and Reflexivity of Marked Poset Polytopes

“…Their motivating example is the Gelfand-Tsetlin polytopes and the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytopes [11], which are respectively marked order polytopes and marked chain polytopes associated to particular marked posets (which are in fact distributive lattices). The toric varieties associated to these polytopes can be obtained from toric degenerations of flag varieties of type A ( [14,20,10]). The geometric properties of the toric varieties associated to Gelfand-Tsetlin polytopes are investigated in [3].…”

On toric degenerations of flag varieties

“…This approach is close to the one which has been used in [27,30,31,32]. It turns out that for a fixed sequence of positive roots and fixed choice of the total order (see section 5.6):…”

“…The procedure uses representation theory as well as ideas from the Newton-Okounkov theory. The strategy can be seen as a common generalization of Caldero's degeneration (and the subsequent constructions of flat toric degenerations of flag varieties by Alexeev-Brion [1] and Kaveh [52]) and the construction of flat toric degenerations using a refinement of the PBW-filtration [32] (see also [30,31]). We will see that both: Caldero's construction and the construction via filtrations have implicitly in common the idea to replace the group U − by a birationally equivalent product of root subgroups.…”

“…In particular, the degenerations of flag varieties by Caldero [10] and Alexeev and Brion [1], as well as the degenerations via filtrations [32] show up in a uniform framework.…”

A Continuous Family of Marked Poset Polytopes

“…The Hibi-Li conjecture has the following geometric application. There exists a flat degeneration of the full flag variety to the toric variety associated to the Gelfand-Tsetlin polytope [11], and also a flat degeneration to the toric variety associated to its marked chain counterpart [5]. The Hibi-Li conjecture would allow a quantitative comparison of the two toric degenerations, for example a comparison of the number of torus fixed points.…”