February 21, Wednesday, 15:30, room 211

Ekaterina Melikhova

Grassmannians and Littlewood-Richardson rule

The complex Grassmanian G_{m,n} is the space of linear subspaces of dimension m in the complex (m+n)-dimensional vector space. Last time for each partition contained in the m by n rectangle, Anastasiya defined the corresponding Schubert cell as a certain subset of G_{m,n}. This time we discuss that Schubert cells form a cellular decomposition of the Grassmanian and that the Poincar\’e duals of the fundamental classes of their closures (Schubert classes) form a basis of the integral cohomology of G_{m,n}. The main objective of the talk is to describe the cup-product in this ring. To this end we describe an epimorphism from the ring of symmetric polynomials in m variables with integer coefficients to the integral cohomology ring of G_{m,n}. This epimorphism sends the Schur polynomial corresponding to a partition \lambda to the Schubert class corresponding to \lambda, as long as \lambda contained in the m by n rectangle and otherwise to zero. The Littlewood–Richardson rule enables one to represent the product of two Schur polynomials as a linear combination of Schur polynomials and a similar rule works for Schubert classes due to the said epimorphism. We give a combinatorial description of this rule via Young tableaux.