Connected components of real double Bruhat cells

TL;DR

This paper determines the number of connected components of real double Bruhat cells in semisimple groups, advancing understanding of their topology and confirming a prior conjecture, with implications for total positivity and Poisson-Lie groups.

Contribution

It extends previous results to all semisimple groups, proving a conjecture about the connected components of real double Bruhat cells.

Findings

Enumeration of connected components in general semisimple groups

Extension of previous results from simply-laced to all cases

Proof of a conjecture related to the topology of these cells

Abstract

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.