Báo cáo toán học: "Affine partitions and affine Grassmannians"

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: Affine partitions and affine Grassmannians. | Affine partitions and affine Grassmannians Sara C. Billey Department of Mathematics University of Washington Seattle WA billey@ Stephen A. Mitchell Department of Mathematics University of Washington Seattle WA mitchell@ Submitted Mar 25 2008 Accepted Jun 24 2009 Published Jul 2 2009 Mathematics Subject Classification 05E15 14M15 Abstract We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott s formula these colored partitions give rise to some partition identities. In certain types these identities have previously appeared in the work of Bousquet-Melou-Eriksson Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type An the affine colored partitions form another family of combinatorial objects in bijection with n 1 -core partitions and n-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmanni-ans in terms of affine partitions and a generalization of Young s lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted. 1 Introduction Let W be a finitejrreducible Weyl group associated to a simple connected compact Lie group G and let W be its associated affine Wey group. In analogy with the Grassmannian manifolds in classical type A the quotient W W is the indexing set for the Schubert varieties injhe affine Grassmannians LG. Let Ws be the minimal length coset representatives for W W. Much of the geometry and topology for the affine Grassmannians can be studied from the combinatorics of Ws and vice versa. For example Bott 6 showed that the Poincare series for the cohomology ring for the affine Grassmannian is equal to . was supported by UW Royalty Research Grant. . was supported by the National Science Foundation. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R18 1 the length .

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