Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and\n flat degenerations

2013/06/06 by Evgeny Feigin, Feigin, Evgeny, Ghislain Fourier +3 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1306.1292

openalex publication_date 2013/06/06 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We introduce the notion of a favourable module for a complex unipotent\nalgebraic group, whose properties are governed by the combinatorics of an\nassociated polytope. We describe two filtrations of the module, one given by\nthe total degree on the PBW basis of the corresponding Lie algebra, the other\nby fixing a homogeneous monomial order on the PBW basis.\n In the favourable case a basis of the module is parameterized by the lattice\npoints of a normal polytope. The filtrations induce flat degenerations of the\ncorresponding flag variety to its abelianized version and to a toric variety,\nthe special fibres of the degenerations being projectively normal and\narithmetically Cohen-Macaulay. The polytope itself can be recovered as a\nNewton-Okounkov body. We conclude the paper by giving classes of examples for\nfavourable modules.\n

Citations

• PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams

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• Extremal part of the PBW-filtration and E-polynomials
• The induced PBW filtration, Frobenius splitting of double flag\n varieties, and Wahl's conjecture

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