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Báo cáo toán học: "Equidimensionality of the Brauer loop scheme"

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: Equidimensionality of the Brauer loop scheme. | Equidimensionality of the Brauer loop scheme Brian Rothbach University of California Berkeley rothbach@math.berkeley.edu Submitted Aug 5 2009 Accepted May 11 2010 Published May 20 2010 Mathematics Subject Classification 14M99 Abstract We give another description of certain subvarieties of the Brauer loop scheme of Knutson and Zinn-Justin. As a consequence we show that the Brauer loop scheme is equidimensional. Contents 1 Introduction 1 2 A decomposition and the dimension of the Fn s 3 3 A geometric description 4 4 A cyclic action on the Fn 8 5 Equations for the Fn s 9 1 Introduction Let N be a positive integer. An integer sequence i1 . ik G 1 . N k is said to be cyclically ordered if either i1 i2 ik or i1 ik and for some 1 c l c k the cyclically rotated sequence il il 1 . ik i1 . il-1 is weakly increasing. We will write o i1 . ik as shorthand for the statement the sequence i1 . ik is cyclically ordered . Knutson and Zinn-Justin 1 defined a nonstandard multiplication on MN C the set of N X N complex matrices by setting P Q ik y PijQjk. We refer to their j O i j k paper as a reference for several nice geometric models of this multiplication. We recall the following facts from their paper. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R75 1 1. A matrix M is invertible under if and only if the diagonal entries are nonzero. The set of invertible matrices under is a solvable Lie group with the invertible diagonal matrices T serving as a maximal torus and with unipotent radical U the set of all matrices with ones along the diagonal. 2. Let E M M 0 M G MN C which can be described set theoretically by the possibly nonreduced equations M M ij 0 for 1 i j N and Mu 0 for 1 i N. Then E n Fn ng I where I c SN is the set of involutions in Sn and for each n G I Fn is the set of all matrices M G E such that the upper triangular part of M is Borel conjugate to the strictly upper triangular part of n. 3. Each Fn is a union of U orbits in other words U Fn Fn. 4. Suppose n has k .