tailieunhanh - Đề tài " A geometric LittlewoodRichardson 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],. | Annals of Mathematics A geometric Littlewood- Richardson rule By Ravi Vakil Annals of Mathematics 164 2006 371 422 A geometric Littlewood-Richardson rule By Ravi Vakil 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 Held 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 Hrst 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 References Appendix A. The bijection between checkergames and puzzles with A. Knutson Appendix B. Combinatorial summary of the rule 1. Introduction A Littlewood-Richardson rule is a combinatorial interpretation of the Littlewood-Richardson numbers. These numbers have a variety of interpre Partially supported by NSF Grant DMS-0228011 an AMS Centennial Fellowship and an Alfred P. Sloan Research Fellowship. 372 RAVI VAKIL tations most often in terms of symmetric functions representation theory and geometry. In each case they appear as structure coefficients of rings. For example in the ring of symmetric functions they are the structure coefficients with respect to the basis of Schur polynomials. In geometry Littlewood-Richardson numbers are structure coefficients of the cohomology ring of the Grassmannian with respect to

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