## Parametrizations of flag varieties

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- by B. R. Marsh and K. Rietsch
- Represent. Theory
**8**(2004), 212-242 - DOI: https://doi.org/10.1090/S1088-4165-04-00230-4
- Published electronically: May 26, 2004

## 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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## Bibliographic 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