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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: The ladder crystal. | The ladder crystal Chris Berg Fields Institute Toronto ON Canada cberg@fields.utoronto.edu Submitted Jan 21 2010 Accepted Jun 28 2010 Published Jul 10 2010 Mathematics Subject Classifications 05E10 20C08 Abstract In this paper I introduce a new description of the crystal B A0 of sfi. As in the Misra-Miwa model of B Ao the nodes of this crystal are indexed by partitions and the i-arrows correspond to adding a box of residue i. I then show that the two models are equivalent by interpreting the operation of regularization introduced by James as a crystal isomorphism. 1 Introduction The main goal of this paper is to give a combinatorial description of the crystal of the basic representation of sic Misra and Miwa previously gave such a description which involved -regular partitions and which I will denote as regg. My description denoted laddg satisfies the following properties The nodes of laddt are partitions and there is an i-arrow from A to ụ only when the difference ụ A is a box of residue i. regt laddt and this crystal isomorphism yields an interesting bijection on the nodes. The map being used for the isomorphism has been well studied 1 but never b efore in the context of a crystal isomorphism. The partitions which are nodes of laddt can be identified by a simple combinatorial condition. 1.1 Background and Previous Results Let A be a partition of n written A r n and I 3 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. P will denote the set of all partitions. An -regular partition is one THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R97 1 in which no part occurs I or more times. To each box x y in a Young diagram of A the residue of that box is the difference y x taken modulo T For two partitions A and y of n we say that A y if Zj i Aj c Zj i yj for all i. This order is usually called the dominance order. The hook length of the a c box of A is defined to be the number of boxes
