Luis Serrano

Symmetric functions and how they appear everywhere

This talk is an appetizer for Max's talk next week about the cohomology of the Grassmannian.

Symmetric functions are polynomials in infinite variables, which are invariant under the action of permuting the variables. They form a ring, which has the Schur functions as an important (vector space) basis. Their product coefficients are the Littlewood-Richardson numbers. Schur functions and the Littlewood-Richardson numbers appear everywhere. Some examples are the following:

* in enumerative geometry questions

* when calculating the cohomology of the Grassmannian

* in the representations of the symmetric group, and the special and linear groups

* in determining the eigenvalues of the sum of two hermitian matrices

In this talk, we will study the structure of the ring of symmetric functions, and show a combinatorial interpretation for the Littlewood-Richardson numbers. This interpretation involves a way to multiply tableaux which might remind you of a certain childhood game. If time permits we might touch on Schubert polynomials, but that might be a pipe dream.