One-element commutation classes
Bridget Eileen Tenner
本文刻画了任意排列中属于自身换位类的既约字，并证明了当排列为长元素n(n-1)...321且n≥4时恰好存在四个这样的字。
对于任意排列w，我们刻画了w中属于其自身换位类的既约字。当w为长元素n(n-1)...321且n≥4时，恰好有四个这样的字。
For any permutation w, we characterize the reduced words of w that are their own commutation class. When w is the long element n(n-1)...321 and n \ge 4, there are exactly four such words.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Maximum number of one-element commutation classes of a permutation
本文研究了排列的单元素交换类的最大数目，并通过上界证明了缩减词数目与交换类数目的关系猜想。
Reduced decompositions and commutation classes
我们研究了约化分解的交换类的三个方面：交换类的数目，它们对应的图的结构，以及子网络的计数，这是Warrington最近引入的一个概念[21]。我们对对易类个数的界推广了Knuth[12]、Green和Losonczy [7]以及Tenner [19]的工作。我们使用模式避免来分析图G（w）的结构，这提供了Tenner在[19]中对vexillary置换的表征的应用。我们还讨论了我们的工作和最近的事态发展之间的联系，在强Bruhat秩序和更高的Bruhat秩序。
We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.
We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and maximum. This characterization also allows us to compute the diameter of the commutation graph for any permutation, from which the results for the longest permutation and for fully commutative permutations are recovered.
Word posets, complexity, and Coxeter groups
由符号的集合$S$生成的幺半群$M$可以被描述为在某些关系下$S$中有限字的等价类的集合，这些关系指定了符号的某个连续序列何时可以被另一个符号序列替换。如果是$a,b\in S$，则形式为$ab=ba$的关系称为\emph{commutation relation}。仅使用一系列对易关系而等价的字被称为在相同的\emph{commutation class}中。我们引入某些偏序集，我们称之为\emph{word posets}，捕获的结构，在幺半群的交换类。词偏序集的同构类与交换类是双射对应的，并且我们证明了词偏序集的线性扩张与交换类中的词是双射对应的，从而证明了在幺半群的交换类中枚举词是P-完全的.然后，我们将单词偏序集应用于Coxeter群。证明了Coxeter群元素约化字的计数问题是P-完全的。我们还演示了一种方法，找到一个元素的约化字的交换类的字偏序集，然后使用这个找到约化字的交换类的数量的递归公式。
Commutation classes of reduced words and higher Bruhat orders for affine permutations
本文研究了仿射排列中交换类的简化解和更高布里哈特序的结构，提出了一种推广弱序区间[ id, w ]的通用方法，揭示了交换类与布里哈特序之间的关系。
On the number of commutation classes of the longest element in the symmetric group
本文研究对称群中最长元素的交换类计数问题，利用标准 Coxeter 表示定义交换等价，并探讨其数量。
Commutation classes of double wiring diagrams
本文提出了一种计算双接线图换流类图的新方法。使用这些方法，我们计算的图形为五个字符串或更少，这使我们能够确认一个积极的猜想Fomin和Zelevinsky时，n是小于五。
Complexes of Injective Words and Their Commutation Classes
设$S$为有限字母表。在$S$上的内射词是在$S$上的词，使得$S$中的每个字母在该词中最多出现一次。研究了$S$上单射词的布尔胞腔复形及其对易类。这概括了法默、比约纳和瓦克斯关于所有内射词的复合体的工作。
We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion o…
We compute the expected number of commutations appearing in a reduced word for the longest element in the symmetric group. The asymptotic behavior of this value is analyzed and shown to approach the length of the permutation, meaning that nearly all positions in the reduced word are expected to support commutations. Finally, we calculate the asymptotic frequencies of commutations, consecutive noncommuting pairs, and long braid moves.
On Graphs of Sets of Reduced Words
本文研究了有限对称群中既约字集合的图结构，提出了子图结构的新研究及计算辫边与交换边数量的新公式，并探讨了辫类和交换类数量的界限。
We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our previous work for vexillary permutations. We also analyze general tilings of Elnitsky's polygon, and demonstrate that these are closely related to the patterns in a permutation. Building on previous work for commutation classes, we show that reduced word enumeration is monotonically increasing with respect to pattern containment. Finally, we give several appl…
Conjugacy of Coxeter elements
对于Coxeter群（W，S），集合S的置换称为Coxeter字，由乘积表示的群元素称为Coxeter元素。将第一个字母移动到单词的末尾称为旋转，如果两个Coxeter元素的单词可以通过一系列旋转和合法交换相互转换，则它们是旋转等价的。我们证明了Coxeter元是共轭的当且仅当它们是旋转等价的。这在某些特殊情况下是已知的，但在一般情况下不适用于Coxeter群。
In this paper, we introduce the notion of combinatorial Auslander-Reiten(AR) quiver for commutation classes $[\widetilde{w}]$ of $w$ in finite Weyl group. This combinatorial object visualizes the convex partial order $\prec_{[\widetilde{w}]}$ on the subset $Φ(w)$ of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection maps, we can apply their properties to the representation theory of KLR algebras and multiplication structure of dual PBW generators associated to any commutation class $[\widetilde{w}_0]$ of the longest element $w_0$.
Reduced Decompositions and Permutation Patterns
Billey、Jockusch和Stanley用约化分解的性质刻画了321-避免置换。本文通过对置换的简化分解和模式包容的概念进行了详细的研究，从而推广了这一结果。这些技巧被用来证明vexillary置换的一个新的特征，在其主对偶阶理想在一个特定的偏序集。此外，组合框架产生了几个新的结果置换的交换类。特别是，这些描述的结构方面的相应的图形的类和zonotopal平铺的多边形定义的Elnitsky是与置换。
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric group which encodes the character of a certain irreducible representation of a Lie group of type $A$. In this paper, we provide a recursive formula for the number of reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand-Cetlin polytope. The recursive formula involves the number of standard Young tableaux…
We consider permutation classes having two basis elements of size three and one further basis element. We completely classify the possible enumeration sequences of such classes and demonstrate that there are far fewer of them than might be expected in principle.
Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's Formula, the method of Lagrange multipliers, and the singular value decomposition of matrices.
In a Coxeter group $W$, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutation of adjacent letters. These elements have particularly nice combinatorial properties, and also index a basis of the generalized Temperley--Lieb algebra attached to $W$. We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that it always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify …
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-m…
The conflated expression graph for an arbitrary permutation
本文证明了任意排列的混合表达式图存在唯一最小元和最大元，且每个约化表达式位于从源到汇的极大链上，推广了Manin-Schechtman关于高阶Bruhat序的工作，并给出了Hothem中某些结果的独立自包含证明。
Symmetry classes connected with the magnetic Heisenberg ring
本文定义了一维自旋为1/2的海森堡磁环的希尔伯特空间H中的对称类和对易对称，并利用对称群S_N的表示论中的工具，如群环C[S_N]的理想分解、C[S_N]的幂等元、S_N的离散傅里叶变换、Littlewood-Richardson积等，研究了它们.特别是，我们确定最小的对称类和稳定子群的单个本征向量v和子空间U的本征向量的哈密顿的磁铁。确定U的最小对称类是基于一个算法，该算法显式地计算C[S_N]的右理想的非直和的生成幂等元。设U是权为（r_1，r_2）的Hamilton算子的固定特征值μ的特征向量的子空间.如果确定U中每个v的最小对称类，则可以观察到对称行为的跳跃。对于“generic "v，所有最小的对称类都有相同的最大维数d和结构。但是U可以包含线性子空间，在该线性子空间上，v的最小对称类的维数跳到小于d的值。则v的稳定子群可以增加。我们可以精确地计算这种跳跃。在我们的调查中，我们使用计算机计算的数学软件包PERMS和HRing。
Counting the number of elements in the mutation classes of \tilde{A}_n-quivers
本文证明了\tilde{A}_n型非同构簇倾斜代数在导出等价类中的个数的显式公式。特别是，我们得到了\tilde{A}_n型箭盒的突变类中的元素数。作为一个副产品，这为Buan和Torkildsen首先确定的Dynkin型箭图D_n的个数提供了另一种证明。
On Enumeration of Conjugacy Classes of Coxeter Elements
本文研究了图Y的无圈定向集上的等价关系，该定向集是由图Y的source-to-sink转换而产生的。这种源-汇转换编码例如Coxeter群的Coxeter元素的共辄。本文利用非桥边的边删除和边收缩，给出了任意图Y的这个关系的等价类数的一个递归的直接证明。最后，我们展示了如何通过Tutte多项式作为T（Y，1，0）的评估也可以获得这个结果，我们提供了双射到其他两类已知以相同方式计数的非循环方向。给出了等价类集合的一个横截。
Counting Conjugacy Classes of Elements of Finite Order in Lie Groups
利用组合技巧，我们回答了两个关于简单经典李群的问题。定义$N(G,m)$为李群$G$中有穷级$m$的元素的共轭类的个数，$N(G,m,s)$为元素具有$s$不同特征值或特征值共轭对的类的个数。什么是$N(G,m)$？对于$G$，是酉群、正交群还是辛群？什么是适用于这些群体的$N(G,m,s)$？对于某些情况，第一个问题在几十年前就通过群论技术得到了回答。看来第二个问题以前没有人问过，这里它是受到弦理论中真空计数问题的启发。我们的组合方法使我们能够回答这两个问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。