Double Bruhat cells and total positivity

Abstract

We study the totally nonnegative variety G ≥ 0 G_{\ge 0} in a semisimple algebraic group G 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 ≥ 0 G_{\ge 0} with double Bruhat cells (intersections of cells of the two Bruhat decompositions of G G with respect to opposite Borel subgroups).