A geometric approach to the combinatorics of Schubert polynomials

Schubert polynomials, which had their origins in the cohomology of flag varieties, have recently been the subject of much interest in algebraic combinatorics. A basic open problem is to give a rule for multiplying two Schubert polynomials, that is, find an analog of the Littlewood-Richardson rule for Schubert polynomials. We describe a geometric proof of an analog of Pieri's rule for Schubert polynomials. This was stated by Lascoux and Schützenberger, who suggested an algebraic proof. Interpreting this formula geometrically vexposes a striking link to the Littlewood-Richardson rule for Schur polynomials, and indicates possible extensions. While this approach uses ideas and methods from algebraic geometry, the proofs involve little more than elementary linear algebra.