Connected components of real double Bruhat cells

Details Connected components of real double Bruhat cells Connected components of real double Bruhat cells

2000-03-31

arxiv.org/abs/math/0003231v2

Mathematics

Algebraic Geometry

18 pages, 4 figures; the title corrected, the abstract slightly modified, one reference added, the proof of Lemma 4.4 simplifi

Scientific paper

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they are also closely related to symplectic leaves in the corresponding Poisson-Lie groups. The term "cells" might be misleading because their topology can be quite non-trivial. As a first step towards understanding this topology, we enumerate the connected components of real double Bruhat cells. This result extends (from the simply-laced case to the general one) and proves the conjecture made in a joint work with B.Shapiro-M.Shapiro-A.Vainshtein; it also extends earlier work by B.Shapiro-M.Shapiro-A.Vainshtein and K.Rietsch.

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