Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups
Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups
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多边形的菱形平铺和 Coxeter 群中的简化词类
DOI：
10.1006/jcta.1997.2723
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
S. Elnitsky
中科院分区：
文献类型：
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作者：
S. Elnitsky
在标准的Coxeter表示中，由相邻的转置（1，2），（2，3），？，（n？1，n）。对于任何给定的置换，我们将其所有最小长度因子分解视为生成元的积。任何两个转置（i，i+1）和（j，j+1）交换，如果数si和j不连续;因此，在任何因式分解中，它们的顺序可以交换，以获得相同置换的另一个因式分解。扩展到一个等价关系，我们建立了一个双射之间的等价类和菱形平铺的某个2n边形确定的置换。我们还研究了由其他Coxeter关系在平铺集上诱导的图结构。对于一个特殊的情形，我们利用格路图证明了Kuperberg和Propp的一个计数猜想及其aq-类似物。最后，我们给出了类似的结构的其他两个家庭的有限Coxeter群，即那些类型BandD。
In the standard Coxeter presentation, the symmetric groupSnis generated by the adjacent transpositions (1, 2), (2, 3), ?, (n?1, n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i, i+1) and (j, j+1) commute if the numbersiandjare not consecutive; thus, in any factorization, their order can be switched to obtain another factorization of the same permutation. Extending this to an equivalence relation, we establish a bijection between the resulting equivalence classes and rhombic tilings of a certain 2n-gon determined by the permutation. We also study the graph structure induced on the set of tilings by the other Coxeter relations. For a special case, we use lattice-path diagrams to prove an enumerative conjecture by Kuperberg and Propp, as well as aq-analogue thereof. Finally, we give similar constructions for two other families of finite Coxeter groups, namely those of typesBandD.