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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Minuscule Heaps Over Dynkin diagrams ˜ of type A. | Minuscule Heaps Over Dynkin diagrams of type A Manabu HAGIWARA Institute of Industrial Science University of Tokyo e-mail manau@imailab.iis.u-tokyo.ac.jp Submitted Oct 18 2002 Accepted Dec 12 2003 Published Jan 2 2004 Abstract A minuscule heap is a partially ordered set together with a labeling of its elements by the nodes of a Dynkin diagram satisfying certain conditions derived by J. Stembridge. This paper classifies the minuscule heaps over the Dynkin diagram of type Ã. 1 Introduction The aim of this paper is to classify the minuscule heaps over a Dynkin diagram of type Ji. Let A be a symmetrizable generalized Cartan matrix and let g be a corresponding Kac-Moody Lie algebra. Let r be a Dynkin diagram which is an encoding of A. Minuscule heaps arose in connection with A-minuscule elements of the Weyl group W of g. According to Proctor 9 and Stembridge 12 the notion of A-minuscule elements of W was defined by Peterson in his unpublished work in the 1980 s. For an integral weight A an element w of W is said to be a A-minuscule element if it has a reduced decomposition Si1 si2 . . . sir such that Sij Sij 1 .Sir A Sij 1 .Sir A - aij 1 yj r and it is called minuscule if w is A-minuscule for some integral weight A. Here ai is the simple root corresponding to si. It is known that a minuscule element is fully commutative namely any reduced decomposition can be converted into any other by exchanging adjacent commuting generators several times see 9 15 10 Theorem A and 11 Theorem 2.2 or 12 Proposition 2.1 . To a fully commutative element w one can associate a r-labeled poset called its heap. A r-labeled poset is a triple P ộ in which P is a poset and ộ P N r is any map called the labeling map . A linear extension of a r-labeled poset naturally The author s name is HAGIWARA Manabu by Japanese ordering. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R3 1 determines a word in the generators of W. The heap of the fully commutative element w is a T-labeled poset whose linear .
