TAILIEUCHUNG - Báo cáo toán học: "The complete cd-index of dihedral and universal Coxeter groups"

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: The complete cd-index of dihedral and universal Coxeter groups. | The complete cd-index of dihedral and universal Coxeter groups Saul A. Blanco Department of Mathematics Cornell University Ithaca NY 14853 UsA sabr@ Submitted Jun 7 2011 Accepted Aug 15 2011 Published Sep 2 2011 Mathematics Subject Classifications 05E99 20F55 05E15 Abstract We present a description including a characterization of the complete cd-index of dihedral intervals. Furthermore we describe a method to compute the complete cd-index of intervals in universal Coxeter groups. To obtain such descriptions we consider Bruhat intervals for which Bjorner and Wachs s CL-labeling can be extended to paths of different lengths in the Bruhat graph. While such an extension cannot be defined for all Bruhat intervals it can be in dihedral and universal Coxeter systems. 1 Introduction Let W S be a Coxeter system and u v E W with u v in Bruhat order. The Bruhat interval u v has been shown to be lexicographically shellable. This was proved by Bjorner and Wachs 3 and Dyer 11 in which the authors construct a CL-labeling and EL-labeling respectively. Dyer s labeling can be given to u-v paths of any length in the Bruhat graph B u v of u v . However Bjorner and Wachs s labeling can only be used in general for the maximal-length paths. Nevertheless there are examples of Bruhat intervals in which both labeling procedures can be used for all u-v paths. We call such intervals BW-labelable for example intervals in dihedral and universal Coxeter systems are BW-labelable. We show that both labelings when defined have the same descentset distribution. Thus one can compute the complete cd-index of Billera and Brenti 1 for such intervals utilizing the BW-labeling. The computation of the complete cd-index for intervals in universal Coxeter groups cannot be carried out in an easy way if at all Partially supported by NSF grant DMS-0555268 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P174 1 utilizing reflection orders as these orders are not easy to generate for infinite .

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