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 .