A Littlewood-Richardson rule for Grassmannian Schubert problems

Kevin Purbhoo and Frank Sottile
    We give a combinatorial rule for the intersection number of a product of Schubert classes pulled back from Grassmannian projections. This rule shows that the intersection number is the number of certain combinatorial objects which we call filtered tableaux. These are sequences of skew Littlewood-Richardson tableaux which 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.
The 18 figures to the left are filtered tableaux which show that the product
p*S2,    p*S2,    p*S3,    p*S3,    p*S4,    p*S4,    p*S5,   
is equal to 18 times the class of a point in the manifold of flags consisting of a 2-plane in a 3-plane in a 5-plane in 7 space.
This picture shows a way to calculate intersection number
( p*S2, )4 ( p*S3, )5 ( p*S4, )4    =    262 [pt]
Multiplying codimension 1 Schubert classes (like these) can be interpreted as counting special chains in the Bruhat order. This "chain-counting" formula is implicit in Monk's rule [Monk]. It is also a special case of this new formula.
[Monk] D. Monk, The geometry of flag manifolds, Proc. London Math. Soc., 9, (1959), pp. 253--286.
[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: 13 May 2007 by Frank Sottile