A geometric Littlewood-Richardson rule

• 成果类型: Article
• 署名作者: Vakil, Ravi
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2006.164.371
• 发表日期: 2006
• 页码: 371-422
• 关键词: enumerative geometry; projective-space; grassmannians; real; COHOMOLOGY; puzzles; CURVES

摘要

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.