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Total positivity, Schubert positivity, and Geometric Satake

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Total positivity, Schubert positivity, and Geometric Satake. / Lam, Thomas; Rietsch, Konstanze.

In: Journal of Algebra, Vol. 460, 15.08.2016, p. 284-319.

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Harvard

Lam, T & Rietsch, K 2016, 'Total positivity, Schubert positivity, and Geometric Satake', Journal of Algebra, vol. 460, pp. 284-319. https://doi.org/10.1016/j.jalgebra.2016.03.039

APA

Lam, T., & Rietsch, K. (2016). Total positivity, Schubert positivity, and Geometric Satake. Journal of Algebra, 460, 284-319. https://doi.org/10.1016/j.jalgebra.2016.03.039

Vancouver

Lam T, Rietsch K. Total positivity, Schubert positivity, and Geometric Satake. Journal of Algebra. 2016 Aug 15;460:284-319. https://doi.org/10.1016/j.jalgebra.2016.03.039

Author

Lam, Thomas ; Rietsch, Konstanze. / Total positivity, Schubert positivity, and Geometric Satake. In: Journal of Algebra. 2016 ; Vol. 460. pp. 284-319.

Bibtex Download

@article{075d742661c54e75b7c260878ecc8635,
title = "Total positivity, Schubert positivity, and Geometric Satake",
abstract = "Let G be a simple and simply-connected complex algebraic group, and let X in the Langlands dual group G^ be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P.The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author.In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence. ",
author = "Thomas Lam and Konstanze Rietsch",
year = "2016",
month = aug,
day = "15",
doi = "10.1016/j.jalgebra.2016.03.039",
language = "English",
volume = "460",
pages = "284--319",
journal = "Journal of Algebra",
issn = "0021-8693",
publisher = "ACADEMIC PRESS INC",

}

RIS (suitable for import to EndNote) Download

TY - JOUR

T1 - Total positivity, Schubert positivity, and Geometric Satake

AU - Lam, Thomas

AU - Rietsch, Konstanze

PY - 2016/8/15

Y1 - 2016/8/15

N2 - Let G be a simple and simply-connected complex algebraic group, and let X in the Langlands dual group G^ be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P.The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author.In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence.

AB - Let G be a simple and simply-connected complex algebraic group, and let X in the Langlands dual group G^ be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P.The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author.In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence.

U2 - 10.1016/j.jalgebra.2016.03.039

DO - 10.1016/j.jalgebra.2016.03.039

M3 - Article

VL - 460

SP - 284

EP - 319

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -

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