TAILIEUCHUNG - Báo cáo toán học: "Product and puzzle formulae for GLn Belkale-Kumar coefficients"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Product and puzzle formulae for GLn Belkale-Kumar coefficients. | Product and puzzle formulae for GLn Belkale-Kumar coefficients Allen Knutson Department of Mathematics Cornell University Ithaca New York USA allenk@ Kevin Purbhoo Combinatorics Optimization Department University of Waterloo Waterloo Ontario Canada kpurbhoo@ Submitted Oct 1 2010 Accepted Mar 24 2011 Published Mar 31 2011 Mathematics Subject Classificaion 05E10 Abstract The Belkale-Kumar product on H G P is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G GLn it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for G P a d 1 -step flag manifold each Belkale-Kumar structure constant is a product of 2 Littlewood-Richardson numbers for which there are many formulae available . the puzzles of Knutson-Tao 03 . This refines previously known factorizations into d 1 factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These BK-puzzles are related to extremal honeycombs as in Knutson-Tao-Woodward 04 using this relation we give another proof of Ressayre s result. Finally we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1 they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2. 1 Introduction and statement of results Let 0 k0 ki k2 . kd n be a sequence of natural numbers and Ff ki . kd be the space of partial flags 0 V1 . Vd Cn dim Vi k-i in Cn. AK was supported by NSF grant 0303523. 1KP was supported by an NSERC discovery grant. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P76 1 Schubert varieties on F ki . kd are indexed by certain words ơ Ơ1 . ơn on a totally ordered alphabet of size d primarily we will use 1 2 . d . The content of Ơ is the sequence m n2 . nd where ni is the number of is in Ơ. We associate

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