Eigenvalue problem and a new product in the cohomology of flag varieties.

Abstract: (Joint work with Shrawan Kumar) Motivated by Horn's conjecture and Eigenvalue problems (eg. possible eigenvalues of products of unitary matrices and saturation conjectures, which I will review), we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P. This product is a certain deformation of the classical product.

This new product is then used to give a more efficient solution of the eigenvalue problem and the problem of determining the existence of G-invariants in the tensor product of irreducible representations of G (this comes from a study of optimally destabilising one parameter subgroups in Geometric Invariant Theory).

If time permits, I will also talk about the relation of this new product to the problem of non-vanishing Schubert structure constants (in a G/P).