# PBW-degenerated Demazure modules and Schubert varieties for triangular elements

@article{Fourier2016PBWdegeneratedDM,
title={PBW-degenerated Demazure modules and Schubert varieties for triangular elements},
author={Ghislain Fourier},
journal={J. Comb. Theory, Ser. A},
year={2016},
volume={139},
pages={132-152}
}
• G. Fourier
• Published 2016
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sl n + 1 . We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements.These faces are again normal polytopes and their Minkowski sum is compatible with tensor… Expand
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#### References

SHOWING 1-10 OF 29 REFERENCES
PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams
• Mathematics
• 2014
We study the PBW filtration on the irreducible highest weight representations of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on theExpand
Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces
• Mathematics
• 2014
We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight.Expand
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
• Mathematics
• 2013
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 ofExpand
Degenerate flag varieties and Schubert varieties: a characteristic free approach
• Mathematics
• 2015
We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for groupExpand
Essential signatures and canonical bases of irreducible representations of the group G2
We consider representations of simple Lie algebras and the problem of constructing a “canonical” weight basis in an arbitrary irreducible finite-dimensional highest-weight module. Vinberg suggested aExpand
Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials
• Mathematics
• 2015
Abstract Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t → ∞ and forExpand
Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration
• Mathematics
• 2014
The Cherednik-Orr conjecture expresses the $t\to\infty$ limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove thisExpand
Extremal part of the PBW-filtration and E-polynomials
• Mathematics
• 2013
Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limitExpand
Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence
We analyze marked poset polytopes and generalize a result due to Hibi and Li, answering whether the marked chain polytope is unimodular equivalent to the marked order polytope. Both polytopes appearExpand
Schubert Geometry of Flag Varieties and Gelfand-Cetlin Theory
This thesis investigates the connection between the geometry of Schubert varieties and Gelfand-Cetlin coordinates on ag manifolds. In particular, we discovered a connection between Schubert calculusExpand