Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Representation Theory
Representation Theory
ISSN 1088-4165

   

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2000): 14M15, 20G20

Retrieve articles in all journals with MSC (2000): 14M15, 20G20


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/S1088-4165-04-00230-4
PII: S 1088-4165(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