TAILIEUCHUNG - Báo cáo toán học: "The ladder crystal"

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@ 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. 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

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