Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: ( , 0)-Carter partitions, their crystal-theoretic behavior and generating function. | 0 -Carter partitions their crystal-theoretic behavior and generating function Chris Berg Department of Mathematics Davis CA 95616 USA berg@ Monica Vaziraniy Department of Mathematics Davis CA 95616 USA vazirani@ Submitted May 16 2008 Accepted Oct 3 2008 Published Oct 13 2008 Mathematics Subject Classifications 05E10 20C30 Abstract In this paper we give an alternate combinatorial description of the 0 -Carter partitions see 4 . The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group see 7 . Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas 7 which is in terms of hook lengths. We use our result to find a generating series which counts such partitions with respect to the statistic of a partition s first part. We then apply our description of these partitions to the crystal graph B Ao of the basic representation of c whose nodes are labeled by -regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all 0 -Carter partitions in the graph B Ao . 1 Introduction Preliminaries Let A be a partition of n and 2 be an integer. We will use the convention x y to denote the box which sits in the xth row and the yth column of the Young diagram of A. Throughout this paper all of our partitions are drawn in English notation. P will denote the set of all partitions. An -regular partition is one in which no nonzero part occurs or more times. The length of a partition A is defined to be the number of nonzero parts of A and is denoted len A . Supported in part by NSF grant DMS-0135345 Supported in part by NSF grant DMS-0301320 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R130 1 The hook length of the a c box of A is defined to be the number of boxes to the right of or below the box a c including the box a c itself. It is denoted h a
