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Parametrizations of flag varieties

Authors: R. J. Marsh and K. Rietsch
Journal: Represent. Theory 8 (2004), 212-242
MSC (2000): Primary 14M15; Secondary 20G20
Published electronically: May 26, 2004
MathSciNet review: 2058727
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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.

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Additional Information

R. J. 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

K. Rietsch
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS

Keywords: Algebraic groups, flag varieties, total positivity, Chamber Ansatz, Deodhar decomposition
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
Article copyright: © Copyright 2004 R.J. Marsh and K. Rietsch