The University of Michigan Combinatorics Seminar


Abstract 

I will describe an explicit geometric LittlewoodRichardson 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 LittlewoodRichardson rules, such as tableaux, and Knutson and Tao's puzzles. This gives the first geometric proof and interpretation of the LittlewoodRichardson rule. The rule has an interpretation in Ktheory, suggested by Buch, which gives an extension of puzzles to Ktheory. I'll also describe the natural extension of this approach to the open question of finding a LittlewoodRichardson 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 LittlewoodRichardson 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.
