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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

   

 

Double Bruhat cells and total positivity


Authors: Sergey Fomin and Andrei Zelevinsky
Journal: J. Amer. Math. Soc. 12 (1999), 335-380
MSC (1991): Primary 22E46; Secondary 05E15, 15A23
MathSciNet review: 1652878
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Abstract: We study the totally nonnegative variety $G_{\ge 0}$ in a semisimple algebraic group $G$. These varieties were introduced by G. Lusztig, and include as a special case the variety of unimodular matrices of a given order whose all minors are nonnegative. The geometric framework for our study is provided by intersecting $G_{\ge 0}$ with double Bruhat cells (intersections of cells of the two Bruhat decompositions of $G$ with respect to opposite Borel subgroups).


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Additional Information

Sergey Fomin
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: fomin@math.mit.edu

Andrei Zelevinsky
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: andrei@neu.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00295-7
PII: S 0894-0347(99)00295-7
Keywords: Total positivity, semisimple groups, Bruhat decomposition
Received by editor(s): February 12, 1998
Additional Notes: The authors were supported in part by NSF grants #DMS-9400914, #DMS-9625511, and #DMS-9700927, and by MSRI (NSF grant #DMS-9022140).
Article copyright: © Copyright 1999 by Sergey Fomin and Andrei Zelevinsky