TAILIEUCHUNG - Báo cáo toán học: " Tight Quotients and Double Quotients in the Bruhat Order"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Tight Quotients and Double Quotients in the Bruhat Order. | Tight Quotients and Double Quotients in the Bruhat Order John R. Stembridge Department of Mathematics University of Michigan Ann Arbor Michigan 48109-1109 USA jrs@ Dedicated to Richard Stanley on the occasion of his 60th birthday Submitted Aug 17 2004 Accepted Jan 31 2005 Published Feb 14 2005 Mathematics Subject Classifications 06A07 20F55 Abstract It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided . double quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases these orbit representations are tight in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order and hence also provides upper bounds for the order dimension. In this paper we 1 provide a general characterization of tightness for one-sided quotients 2 classify all tight one-sided quotients of finite Coxeter groups and 3 classify all tight double quotients of affine Weyl groups. 0. Introduction. The Bruhat orderings of Coxeter groups and their parabolic quotients have a long history that originates with the fact that these posets in the case of finite Weyl groups record the inclusion of cell closures in generalized flag varieties. Some of the significant early papers on the combinatorial aspects of this subject include the 1977 paper of Deodhar D1 providing various characterizations of the Bruhat order including some that will be essential in this work the 1980 paper of Stanley St in which Bruhat orderings of finite Weyl groups and their parabolic quotients are shown to be strongly Sperner .

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