Skip to Main Content

Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Parametrizations of flag varieties
HTML articles powered by AMS MathViewer

by B. R. Marsh and K. Rietsch
Represent. Theory 8 (2004), 212-242
Published electronically: May 26, 2004


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.
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
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:
  • 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:
  • MathSciNet review: 2058727