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@ 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 .
