Looping of the numbers game and the alcoved hypercube

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Gashi, Qendrim R.¹  Schedler, Travis²  Speyer, David E.³ 

[1] Univ Prishtina, FSHMN Dekanati, Prishtine 10000, Kosovo, Serbia

[2] MIT, Cambridge, MA 02139 USA

[3] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

关键词: Mozes's game of numbers;    Weyl groups;    Tits cone;    Hasse diagrams;    Weak and Bruhat orders;   

DOI  :  10.1016/j.jcta.2011.11.012

来源: Elsevier

【 摘 要 】

We study the so-called looping case of Mozes's game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. This strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These posets are self-dual and mutually isomorphic, and their Hasse diagrams are dual to the triangulation of the unit hypercube by reflecting hyperplanes. Unlike the weak and Bruhat orders, the top degree is cubic in the number of vertices of the graph. We explicitly compute the rank generating function of the poset. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free