# 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} }

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 the… Expand

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- 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 of… Expand

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 group… Expand

Essential signatures and canonical bases of irreducible representations of the group G2

- Mathematics
- 2015

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 a… Expand

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 for… Expand

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 this… Expand

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 limit… Expand

Marked poset polytopes: Minkowski sums, indecomposables, and unimodular equivalence

- Mathematics
- 2014

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 appear… Expand

Schubert Geometry of Flag Varieties and Gelfand-Cetlin Theory

- Mathematics
- 2000

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 calculus… Expand