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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Parametrizations of flag varieties
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by B. R. Marsh and K. Rietsch PDF
Represent. Theory 8 (2004), 212-242

Abstract:

For the flag variety $G/B$ of a reductive algebraic group $G$ we define and describe explicitly a certain (set-theoretical) cross-section $\phi : G/B\to G$. The definition of $\phi$ depends only on a choice of reduced expression for the longest element $w_0$ in the Weyl group $W$. It assigns to any $gB$ a representative $g\in G$ together with a factorization into simple root subgroups and simple reflections. The cross-section $\phi$ is continuous along the components of Deodhar’s decomposition of $G/B$. We introduce a generalization of the Chamber Ansatz and give formulas for the factors of $g=\phi (gB)$. These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety $(G/B)_{\ge 0}$ defined by Lusztig, giving a new proof of Lusztig’s conjectured cell decomposition of $(G/B)_{\ge 0}$. We also give minimal sets of inequalities describing these cells.
References
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Additional Information
  • B. R. Marsh
  • Affiliation: Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH
  • Address at time of publication: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH
  • MR Author ID: 614298
  • ORCID: 0000-0002-4268-8937
  • K. Rietsch
  • Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS
  • Email: rietsch@mth.kcl.ac.uk
  • Received by editor(s): February 13, 2004
  • Received by editor(s) in revised form: March 19, 2004
  • Published electronically: May 26, 2004
  • Additional Notes: The first named author was supported by a University of Leicester Research Fund Grant and a Leverhulme Fellowship
    The second named author is supported by a Royal Society Dorothy Hodgkin Research Fellowship
  • © Copyright 2004 B.R. Marsh and K. Rietsch
  • Journal: Represent. Theory 8 (2004), 212-242
  • MSC (2000): Primary 14M15; Secondary 20G20
  • DOI: https://doi.org/10.1090/S1088-4165-04-00230-4
  • MathSciNet review: 2058727