A geometric Littlewood–Richardson rule

Abstract

We describe a geometric Littlewood-Richardson rule, interpreted as deforming the intersection of two Schubert varieties into the union of 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 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. Geometric consequences are described here and in [V2], [KV1], [KV2], [V3]. For example, the rule also has an interpretation in $K$-theory, suggested by Buch, which gives an extension of puzzles to $K$-theory.