Parametrizations of flag varieties
Authors:
R. J. Marsh and K. Rietsch
Journal:
Represent. Theory 8 (2004), 212242
MSC (2000):
Primary 14M15; Secondary 20G20
Published electronically:
May 26, 2004
MathSciNet review:
2058727
Fulltext PDF Free Access
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Abstract: For the flag variety of a reductive algebraic group we define and describe explicitly a certain (settheoretical) crosssection . 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 crosssection 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:
http://dx.doi.org/10.1090/S1088416504002304
PII:
S 10884165(04)002304
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
