Parametrizations of flag varieties

Joint with with K. Rietsch (Kings).

Let G be reductive algebraic group and B a Borel subgroup. The flag
variety is the quotient G/B. We give an explicit description of the
elements of G/B depending on a choice of reduced expression for the
longest element in the Weyl group of G, compatible with the
components
of the Deodhar decomposition of G/B. The factors appearing in the
description are given by a generalisation of the
Berenstein-Zelevinsky
chamber ansatz. We obtain a new proof of Lusztig's conjectured cell
decomposition of the totally nonnegative part of G/B.