A Littlewood-Richardson rule for Grassmannian Schubert problems

Frank Sottile
    The motivating and as yet unsolved question in the modern Schubert calculus is to give a combinatorial formula for the product of Schubert classes in the cohomology ring of a flag manifold. In this talk, I will describe a solution to this problem for a special, but important class of products. These are products of Schubert classes pulled back from Grassmannian projections, and our formula is for the coefficient of the class of a point in such a product. This rule is joint work with Kevin Purbhoo. Our rule shows that this intersection number is the number of certain combinatorial objects we call filtered tableaux, which are sequences of skew Littlewood-Richardson tableaux that together fill a shifted shape.
    Rather than try to describe this rule, we recommend that you read our paper [PS]., which is on the ArXiv at math.CO/0708.1582.
The picture at the left shows the four filtered tableaux corresponding to the degree of the product
p*S1,    p*S2,    p*S3,    p*S4,    p*S5,    =    4 [pt]                
of Schubert classes in the manifold of complete flags in 6-space. Each factor is a Schubert class pulled back from a (in this case, different) Grassmannian, with the first from the Grassmannian of 1-planes, the second from the Grassmannian of 2-planes, and etc.
   The pieces in the filtered tableaux are Littlewood-Richardson skew tableaux corresponding to the partitions that appear in the product. Other than the green parition, , there will always be a unique Littlewood-Richardson skew tableaux (or none) of any given shape. In the one case where there is more than a single Littlewood-Richardson tableau, we have written the two possibilities.
[PS] K. Purbhoo and F. Sottile, A Littlewood-Richardson rule for Grassmannian permutations, ArXiv.org/0708.1582.
Work of Sottile supported by the National Science Foundation under CAREER Grant DMS-0538734.
This work was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation, and by the Centre de recherches mathématiques.
Modified since: 6 November 2007 by Frank Sottile