Remote Access Representation Theory
Green Open Access

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

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?)

  • 1. John C. Baez, Link invariants of finite type and perturbation theory, Lett. Math. Phys. 26 (1992), no. 1, 43-51. MR 93k:57006
  • 2. A. Berenstein and A. Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), 128-166. MR 99g:14064
  • 3. Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49-149. MR 98j:17008
  • 4. Joan S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253-287. MR 94b:57007
  • 5. Armand Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 92d:20001
  • 6. Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499-511. MR 86f:20045
  • 7. S. Fomin and A. Zelevinsky, Double bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335-380. MR 2001f:20097
  • 8. Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press Inc., Boston, MA, 1987. MR 89c:20001
  • 9. David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 81j:20066
  • 10. G. Lusztig, Total positivity in reductive groups, Lie theory and geometry: in honor of Bertram Kostant (G. I. Lehrer, ed.), Progress in Mathematics, vol. 123, Birkhaeuser, Boston, 1994, pp. 531-568. MR 96m:20071
  • 11. R. J. Marsh and K. Rietsch, The intersection of opposed big cells in the real flag variety of type $G\sb 2$, Proc. London Math. Soc. (3) 85 (2002), no. 1, 22-42. MR 2004b:14088
  • 12. K. Rietsch, An algebraic cell decomposition of the nonnegative part of a flag variety, J. Algebra 213 (1999), 144-154. MR 2000e:14086
  • 13. T. A. Springer, Linear algebraic groups, second edition, Progress in Mathematics, vol. 9, Birkhäuser, Boston, 1998. MR 99h:20075
  • 14. Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968, Notes prepared by John Faulkner and Robert Wilson. MR 57:6215

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

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

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

American Mathematical Society