TAILIEUCHUNG - Báo cáo toán học: "Climbing elements in finite Coxeter groups"

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: Climbing elements in finite Coxeter groups. | Climbing elements in finite Coxeter groups Thomas Brady School of Mathematical Sciences Dublin City University Glasnevin Dublin 9 Ireland Aisling Kenny School of Mathematical Sciences Dublin City University Glasnevin Dublin 9 Ireland Colum Watt School of Mathematical Sciences Dublin Institute of Technology Dublin 8 Ireland Submitted May 9 2010 Accepted Nov 8 2010 Published Nov 19 2010 Mathematics Subject Classifications 20F55 05E15 Abstract We define the notion of a climbing element in a finite real reflection group relative to a total order on the reflection set and we characterise these elements in the case where the total order arises from a bipartite Coxeter element. 1 Introduction Suppose W S is a finite Coxeter system. Each reduced expression for an element w of W determines a total order on the inversion set of w. The inversion set of the longest element w0 of W is equal to the set T of all the reflections and a particular reduced expression for w0 gives a total order T on T. For some elements w of W the restriction of T to the inversion set of w coincides with the order determined by one of its reduced expressions. We will call such an element w a climbing element of W. Geometrically this means that there is a gallery from the fundamental domain C to w C which crosses hyperplanes in increasing order. In this paper we characterise the climbing elements in the case where the reduced expression for wo is obtained by iterating a bipartite factorisation of a Coxeter element. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R156 1 This characterisation is obtained using the construction from 6 of a copy of the type-W generalised associahedron whose cone is a coarsening of the fan determined by the W reflection hyperplanes. This coarsening determines an equivalence relation on W whose equivalence classes we prove directly to be intervals in the left weak order. The least elements of these intervals are .

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