Parametrizations of flag varieties

Authors:
R. J. 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

Published electronically:
May 26, 2004

MathSciNet review:
2058727

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Abstract | References | Similar Articles | Additional Information

Abstract: For the flag variety of a reductive algebraic group we define and describe explicitly a certain (set-theoretical) cross-section . The definition of depends only on a choice of reduced expression for the longest element in the Weyl group . It assigns to any a representative together with a factorization into simple root subgroups and simple reflections. The cross-section is continuous along the components of Deodhar's decomposition of . We introduce a generalization of the Chamber Ansatz and give formulas for the factors of . These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety defined by Lusztig, giving a new proof of Lusztig's conjectured cell decomposition of . 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

Email:
rjm25@mcs.le.ac.uk

**K. Rietsch**

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

Email:
rietsch@mth.kcl.ac.uk

DOI:
https://doi.org/10.1090/S1088-4165-04-00230-4

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