Double Bruhat cells and total positivity

Fomin, Sergey; Zelevinsky, Andrei

doi:10.1090/S0894-0347-99-00295-7

Double Bruhat cells and total positivity
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by Sergey Fomin and Andrei Zelevinsky

J. Amer. Math. Soc. 12 (1999), 335-380

DOI: https://doi.org/10.1090/S0894-0347-99-00295-7

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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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