Room 326

An Introduction to Schubert Polynomials

Vijay Ravikumar

Chennai Mathematical Institute

We will go over Lascoux and Schutzenberger's construction of the "Schubert polynomials".  In particular, we will review the Hasse diagram corresponding to the symmetric group S_n, and then introduce the divided difference operators, which allow us to "push" certain polynomials down the Hasse diagram.  Starting with a "nice" polynomial at the very top of the diagram, we will construct the set of Schubert polynomials, one for each element of S_n.  We shall see how these polynomials generalize Schur polynomials, and how they convey important geometric information about the manifold of compete flags on the vector space C^n.  Note that Prof. Takeshi Ikeda will talk about certain generalizations of Schubert polynomials in a later seminar.