2000/03/31 by Andrei Zelevinsky, Zelevinsky, Andrei
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.AG #math.RT
paper · pdf · doi:10.48550/arxiv.math/0003231
18 pages, 4 figures; the title corrected, the abstract slightly modified, one reference added, the proof of Lemma 4.4 simplified
openalex publication_date 2000/03/31 · arxiv created 2000/04/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.