The University of Michigan Combinatorics Seminar
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. This rule should be seen as a generalization of Pieri's rule to arbitrary Schubert classes, by way of explicit homotopies. I will describe the straightforward bijections to other Littlewood-Richardson rules, such as tableaux, and Knutson and Tao's puzzles.
This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. The rule has an interpretation in K-theory, suggested by Buch, which gives an extension of puzzles to K-theory. I'll also describe the natural extension of this 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, including L. Mihalcea's ongoing work on Littlewood-Richardson rules for the symplectic Grassmannian.
There are no restrictions on the base field, and all multiplicities
arising are 1; this is important for applications, some of which will
be described in the algebraic geometry seminar earlier the same week.
Although some geometric aspects of the rule may be discussed in the
algebraic geometry seminar, this talk will be logically independent.