The Western Algebraic Geometry Seminar


Speaker: Ravi Vakil
Title: A geometric Littlewood-Richardson rule
I will describe an explicit geometric Littlewood-Richardson rule,
interpreted as deforming the intersection of two Schubert varieties so
that they break into Schubert varieties. There are no restrictions on
the base field, and all multiplicities arising are 1; this is
important for applications. This rule should be seen as a
generalization of Pieri's rule to arbitrary Schubert classes, by way
of explicit homotopies. It has a straightforward bijection to other
Littlewood-Richardson rules, such as tableaux and Knutson and Tao's

This gives the first geometric proof and interpretation of the
Littlewood-Richardson rule. It has a host of geometric consequences,
which I may describe, time permitting. The rule also has an
interpretation in K-theory, suggested by Buch, which gives an
extension of puzzles to K-theory, and in fact a Littlewood-Richardson
rule in equivariant K-theory (ongoing work with Knutson). The rule
suggests a natural approach to the open question of finding a
Littlewood-Richardson rule for the flag variety, leading to a
conjecture, shown to be true up to dimension 5. Finally, the rule
suggests approaches to similar open problems, such as
Littlewood-Richardson rules for the symplectic Grassmannian and
two-flag varieties.