A geometric Littlewood–Richardson rule

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. Contents 1. Introduction 2. The statement of the rule 3. First applications: Littlewood-Richardson rules 4. Bott-Samelson varieties 5. Proof of the Geometric Littlewood-Richardson rule (Theorem 2.13) References Appendix A. The bijection between checkergames and puzzles (with A. Knutson) Appendix B. Combinatorial summary of the rule