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.


On Mirković-Vilonen cycles and crystal combinatorics
HTML articles powered by AMS MathViewer

by Pierre Baumann and Stéphane Gaussent
Represent. Theory 12 (2008), 83-130
Published electronically: March 5, 2008


Let $G$ be a complex connected reductive group and let $G^{\vee }$ be its Langlands dual. Let us choose a triangular decomposition ${\mathfrak n}^{-,{\vee }} \oplus {\mathfrak h}^{\vee }\oplus {\mathfrak n}^{+,{\vee }}$ of the Lie algebra of $G^{\vee }$. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine Grassmannian $G\bigl (\mathbb C((t))\bigr )/G\bigl (\mathbb C[[t]]\bigr )$ is a crystal isomorphic to the crystal of the canonical basis of $U({\mathfrak n}^{+,{\vee }})$. Starting from the string parameter of an element of the canonical basis, we give an explicit description of a dense subset of the associated MV cycle. As a corollary, we show that the varieties involved in Lusztig’s algebraic-geometric parametrization of the canonical basis are closely related to MV cycles. In addition, we prove that the bijection between LS paths and MV cycles constructed by Gaussent and Littelmann is an isomorphism of crystals.
Similar Articles
Bibliographic Information
  • Pierre Baumann
  • Affiliation: Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
  • Email:
  • Stéphane Gaussent
  • Affiliation: Institut Élie Cartan, Unité Mixte de Recherche 7502, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France
  • Email:
  • Received by editor(s): April 26, 2007
  • Received by editor(s) in revised form: October 17, 2007
  • Published electronically: March 5, 2008
  • Additional Notes: Both authors are members of the European Research Training Network “LieGrits”, contract no. MRTN-CT 2003-505078.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 12 (2008), 83-130
  • MSC (2000): Primary 20G05; Secondary 05E15, 14M15, 17B10, 22E67
  • DOI:
  • MathSciNet review: 2390669