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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: Cyclic sieving for longest reduced words in the hyperoctahedral group. | Cyclic sieving for longest reduced words in the hyperoctahedral group T. Kyle Petersen Luis Serrano Department of Mathematical Sciences Department of Mathematics DePaul University Chicago IL University of Michigan Ann Arbor MI tpeter21@depaul.edu lserrano@umich.edu Submitted Jun 2 2009 Accepted Apr 22 2010 Published Apr 30 2010 Mathematics Subject Classifications 05E10 05E15 05E18 Abstract We show that the set R w0 of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon. More specifically R w0 possesses a natural cyclic action given by moving the first letter of a word to the end and we show that the orbit structure of this action is encoded by the generating function for the major index on R w0 . 1 Introduction and main result Suppose we are given a finite set X a finite cyclic group C u acting on X and a polynomial X q G Z q with integer coefficients. Following Reiner Stanton and White RSW we say that the triple X C X q exhibits the cyclic sieving phenomenon CSP if for every integer d 0 we have that X X Zd where Z G C is a root of unity of multiplicitive order C and X is the fixed point set of the action of the power ud. In particular since the identity element fixes everything in any group action we have that X X 1 whenever X C X q exhibits the CSP. If the triple X C X q exhibits the CSP and Z is a primitive C th root of unity we can determine the cardinalities of the fixed point sets X1 X X X 2 . X 1 via the polynomial evaluations X 1 X Z X Z2 . X Z C -1 . These fixed point set sizes determine the cycle structure of the canonical image of u in the group of permutations of X Sx. Therefore to find the cycle structure of the image of any bijection u X X it is enough to determine the order of the action of u on X and find a polynomial X q such that X u X q exhibits the CSP. The cyclic sieving phenomenon has been demonstrated in a variety of contexts. The paper of Reiner Stanton and White RSW itself includes
