论文标题
在12和21交流的简化单词上的图形直径
Diameters of Graphs on Reduced Words of 12 and 21-Inflations
论文作者
论文摘要
经典的结果是，可以通过一系列相邻的换位序列生成对称组中的任何置换。最小长度的序列称为简化单词，在本文中，我们研究了这些简化单词的图，其边缘由基础Coxeter组的关系确定。最近，直径是根据Reiner和Roichman和Assaf的最长置换$ N \ LDOTS 21 $计算的。在本文中，我们发现了12-充电图和许多21通路图的直径的感应公式。这些结果扩展到有关换向和长辫类的相关图。同样，这些结果给出了最长排列的直径的递归公式，该公式与Reiner，Roichman和Assaf的直径相匹配。最后，我们基于基于潜在的超平面布置的Reiner和Roichman在直径的猜想边界上取得了进展，并找到了实现猜想的上限和下限的排列家族。特别是避免312或231的排列具有实现上限的图形。
It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs on commutation and long braid classes. Also, these results give a recursive formula for the diameter of the longest permutation, which matches that of Reiner, Roichman and Assaf. Lastly, We make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement, and find families of permutations that achieve the upper bound and lower bound of the conjecture. In particular permutations that avoid 312 or 231 have graphs that achieve the upper bound.
