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 Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes. | The Cyclic Sieving Phenomenon for Faces of Cyclic Polytopes Sen-Peng Eu Department of Applied Mathematics National University of Kaohsiung Taiwan 811 . speu@ Tung-Shan FU Mathematics Faculty National Pingtung Institute of Commerce Taiwan 900 . tsfu@ Yeh-Jong Pan Department of Computer Science and Information Engineering Tajen University Taiwan 907 . yjpan@ Submitted Sep 8 2009 Accepted Mar 17 2010 Published Mar 29 2010 Mathematics Subject Classifications 05A15 52B15 Abstract A cyclic polytope of dimension d with n vertices is a convex polytope combinatorially equivalent to the convex hull of n distinct points on a moment curve in Rd. In this paper we prove the cyclic sieving phenomenon introduced by Reiner-Stanton-White for faces of an even-dimensional cyclic polytope under a group action that cyclically translates the vertices. For odd-dimensional cyclic polytopes we enumerate the faces that are invariant under an automorphism that reverses the order of the vertices and an automorphism that interchanges the two end vertices according to the order on the curve. In particular for n d 2 we give instances of the phenomenon under the groups that cyclically translate the odd-positioned and even-positioned vertices respectively. Research partially supported by the National Science Council Taiwan under grant NSC grants 98-2115-M-390-002-MY3 Research partially supported by NSC grants 97-2115-M-251-001-MY2 Research partially supported by NSC grants 98-2115-M-127-001 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R47 1 1 Introduction In 8 Reiner-Stanton-White introduced the following enumerative phenomenon for a set of combinatorial structures under an action of a cyclic group. Let X be a finite set X q a polynomial in Z q with the property X 1 X and C a finite cyclic group acting on X. The triple X X q C is said to exhibit the cyclic sieving phenomenon CSP if for every c G C X q q u x G X c x x 1 where u is a root of .
