The number of connected components in double Bruhat cells for nonsimply-laced groups

Authors:
Michael Gekhtman, Michael Shapiro and Alek Vainshtein

Journal:
Proc. Amer. Math. Soc. **131** (2003), 731-739

MSC (2000):
Primary 20F55; Secondary 05E15, 14M15

Published electronically:
June 12, 2002

MathSciNet review:
1937410

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We compute the number of connected components in a generic real double Bruhat cell for series and and an exceptional group .

**[BFZ]**Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky,*Parametrizations of canonical bases and totally positive matrices*, Adv. Math.**122**(1996), no. 1, 49–149. MR**1405449**, 10.1006/aima.1996.0057**[FZ]**Sergey Fomin and Andrei Zelevinsky,*Double Bruhat cells and total positivity*, J. Amer. Math. Soc.**12**(1999), no. 2, 335–380. MR**1652878**, 10.1090/S0894-0347-99-00295-7**[R1]**Konstanze Rietsch,*Intersections of Bruhat cells in real flag varieties*, Internat. Math. Res. Notices**13**(1997), 623–640. With an appendix by G. Lusztig. MR**1459628**, 10.1155/S107379289700041X**[R2]**Konstanze Rietsch,*The intersection of opposed big cells in real flag varieties*, Proc. Roy. Soc. London Ser. A**453**(1997), no. 1959, 785–791. MR**1447152**, 10.1098/rspa.1997.0043**[SSV97]**B. Shapiro, M. Shapiro, and A. Vainshtein,*Connected components in the intersection of two open opposite Schubert cells in 𝑆𝐿_{𝑛}(𝑅)/𝐵*, Internat. Math. Res. Notices**10**(1997), 469–493. MR**1446839**, 10.1155/S1073792897000329**[SSV98]**B. Shapiro, M. Shapiro, and A. Vainshtein,*Skew-symmetric vanishing lattices and intersections of Schubert cells*, Internat. Math. Res. Notices**11**(1998), 563–588. MR**1631773**, 10.1155/S1073792898000361**[SSV99]**B. Shapiro, M. Shapiro, and A. Vainshtein,*Intersections of Schubert cells and groups generated by symplectic transvections*, Proc. 11th Conf. Formal Power Series and Algebraic Combinatorics (FPSAC'99), 1999, pp. 530-533.**[SSVZ]**Boris Shapiro, Michael Shapiro, Alek Vainshtein, and Andrei Zelevinsky,*Simply laced Coxeter groups and groups generated by symplectic transvections*, Michigan Math. J.**48**(2000), 531–551. Dedicated to William Fulton on the occasion of his 60th birthday. MR**1786504**, 10.1307/mmj/1030132732**[Z]**Andrei Zelevinsky,*Connected components of real double Bruhat cells*, Internat. Math. Res. Notices**21**(2000), 1131–1154. MR**1800992**, 10.1155/S1073792800000568

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20F55,
05E15,
14M15

Retrieve articles in all journals with MSC (2000): 20F55, 05E15, 14M15

Additional Information

**Michael Gekhtman**

Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Email:
Michael.Gekhtman.1@nd.edu

**Michael Shapiro**

Affiliation:
Matematiska Institutionen, KTH, Stockholm, Sweden

Address at time of publication:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
mshapiro@math.kth.se, mshapiro@math.msu.edu

**Alek Vainshtein**

Affiliation:
Departments of Mathematics and of Computer Science, University of Haifa, Israel 31905

Email:
alek@mathcs.haifa.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06604-2

Keywords:
Double Bruhat cells,
Coxeter graphs,
groups generated by transvections

Received by editor(s):
May 8, 2001

Received by editor(s) in revised form:
October 25, 2001

Published electronically:
June 12, 2002

Communicated by:
John R. Stembridge

Article copyright:
© Copyright 2002
American Mathematical Society