Schubert polynomials and Bott-Samelson varieties

Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the D emazure character formula, and the generating. series of semistandard table aux. We produce these missing formulas and obtain several surprising expres sions for Schubert polynomials. The above results arise naturally from a new geometric model of Schubert po lynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott Samelson variety Z as the closure of a B- orbit in a product of Rag varieties. This construction works lot an arbitra ry reductive group G, and for G = GL(n) it realizes Z as the representation s of a certain partially ordered set. This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, whic h are crucial in the combinatorics of canonical bases and matrix factorizat ions. On the other hand, our embedding of Z gives an elementary constructio n of its coordinate ring, and allows us to specify a basis indexed by table aux.