Parametrizations of flag varieties

Abstract

For the flag variety G / B G/B of a reductive algebraic group G G we define and describe explicitly a certain (set-theoretical) cross-section ϕ : G / B → G \phi : G/B\to G . The definition of ϕ \phi depends only on a choice of reduced expression for the longest element w 0 w_0 in the Weyl group W W . It assigns to any g B gB a representative g ∈ G g\in G together with a factorization into simple root subgroups and simple reflections. The cross-section ϕ \phi is continuous along the components of Deodhar’s decomposition of G / B G/B . We introduce a generalization of the Chamber Ansatz and give formulas for the factors of g = ϕ ( g B ) g=\phi (gB) . These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety ( G / B ) ≥ 0 (G/B)_{\ge 0} defined by Lusztig, giving a new proof of Lusztig’s conjectured cell decomposition of ( G / B ) ≥ 0 (G/B)_{\ge 0} . We also give minimal sets of inequalities describing these cells.