Bott-Samelson Varieties and Configuration Spaces

The Bott-Samelson varieties Z are a powerful tool in the representation theory and geometry of a reductive group G. We give a new construction of Z as the closure of a B-orbit in a product of flag varieties (G/B). This also gives an embedding of the projective coordinate ring of the variety into the function ring of a Borel subgroup: C[Z] ⊂ C[B]. In the case of the general linear group G = GL(n), this identifies Z as a configuration variety of multiple flags subject to certain inclusion conditions, controlled by the combinatorics of braid diagrams and generalized Young diagrams. The natural mapping Z → G/B compactifies the matrix factorizations of Berenstein, Fomin and Zelevinsky [2]. As an application, we give a geometric proof of the theorem of Kraskiewicz and Pragacz [12] that Schubert polynomials are characters of Schubert modules. Our work leads on the one hand to a Demazure character formula for Schubert polynomials and other generalized Schur functions, and on the other hand to a Standard Monomial Theory for Bott-Samelson varieties. All our results remain valid in arbitrary characteristic and over Z.