Schubert polynomials and Bott-Samelson varieties

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