Type: Article
Publication Date: 1999-01-01
Citations: 358
DOI: https://doi.org/10.1090/s0894-0347-99-00295-7
We study the totally nonnegative variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a semisimple algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript greater-than-or-equal-to 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">G_{\ge 0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with double Bruhat cells (intersections of cells of the two Bruhat decompositions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with respect to opposite Borel subgroups).