2003/07/02 by Bethany Marsh, Marsh, Bethany, Konstanze Rietsch +1 · 2 citations
Mathematics · #14M15 #20G20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0307017
openalex publication_date 2003/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For the flag variety G/B of a reductive algebraic group G we define a certain (set-theoretical) cross-section phi from G/B to G, which depends on a choice of reduced expression for the longest element in the Weyl group. This cross-section is continuous along the components of Deodhar's decomposition of G/B and assigns to any flag gB a representative phi(gB) in G which comes with a natural factorization into simple root subgroups and simple reflections. We introduce a generalization of the Chamber Ansatz of Berenstein, Fomin and Zelevinsky and use it to obtain formulas for the factors of phi(gB). Our results then allow us parameterize explicitly the components of the totally nonnegative part of the flag variety as defined by Lusztig. This gives a new proof of Lusztig's conjectured cell decomposition of this set.