Joel Brewster Lewis
We show that the Hurwitz action is "as transitive as possible" on reflection factorizations of Coxeter elements in the well-generated complex reflection groups $G(d, 1, n)$ (the group of $d$-colored permutations) and $G(d, d, n)$.
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We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced factorizations into a product of reflections is transitive. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a reduced factorization into a product of reflections that generate a parabolic subgroup.
We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.
We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis and the question whether there is an anlogue of the well known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis and Reiner as well a…
In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants
本文证明有限Coxeter群中抛物型拟Coxeter元素的反射分解的Hurwitz轨道由生成子群和共轭类多重集完全区分，并分类了满足极小生成集为最小大小的有限Coxeter群。
We study the Hurwitz action of the classical braid group on factorisations of a Coxeter element c in a well-generated complex reflection group W. It is well-known that the Hurwitz action is transitive on the set of reduced decompositions of c in reflections. Our main result is a similar property for the primitive factorisations of c, i.e. factorisations with only one factor which is not a reflection. The motivation is the search for a geometric proof of Chapoton's formula for the number of chains of given length in the non-crossing partitions lattice NCP_W. Our proof uses the properties of the…
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W lie as well in this parabolic subgroup.
The Hurwitz Action in the Affine Symmetric Group
本文针对仿射型$\tilde{A}_n$的仿射对称群$\tilde{S}_N$，研究辫群在简约反射分解集合上的Hurwitz作用，确定该作用的所有轨道，解决了有限Coxeter群中Hurwitz作用传递性相关问题的仿射情形。
We prove that two reflection factorizations of a given element in an exceptional rank-2 complex reflection group of tetrahedral type are Hurwitz-equivalent if and only if they generate the same subgroup and have the same multiset of conjugacy classes.
Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
本文在良生成复反射群中定义并刻画抛物拟 Coxeter 元素，计算其约化反射分解与相对生成集的基数，并将后者与 Frobenius 流形几何不变量关联，为后续统一公式奠定基础。
We extend a result of Lewis and Reiner from finite Coxeter groups to all Coxeter groups by showing that two reflection factorizations of a Coxeter element lie in the same Hurwitz orbit if and only if they share the same multiset of conjugacy classes.
We provide a variety of cases in which two factorizations have Hurwitz orbits of the same size. We begin with prototypical results about factorizations of length two, and show that cycling elements or flipping and inverting elements in any factorization preserves Hurwitz orbit size. We prove that "double reverse" factorizations in groups with special presentations have Hurwitz orbits of equal size, and end with applications to complex reflection groups.
Hurwitz Equivalence of Reflection Factorizations in Types $B$ and $D$
本文证明了 Lewis 关于有限 Coxeter 群中反射因子分解的 Hurwitz 等价性猜想在类型 A、B、D 下成立，并给出了将猜想约化为例外类型有限问题的约化方法。
Hurwitz numbers for reflection groups III: Uniform formulas
本研究提出Weyl群或复反射群中抛物拟Coxeter元全反射分解数的统一公式，推广了亏格0 Hurwitz数公式，是三篇系列论文的收官成果。
We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.
In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.
We prove that the Hurwitz action on reflection factorizations of Coxeter elements is transitive up to certain natural constraints in the complex reflection groups G4 and G5. This affirms a more general conjecture by Lewis and Reiner in these specific cases. The proof uses induction on length of the factorization using the fact that the square of a reflection is also a reflection.
We show that in the complex reflection group $G_6$, reflection factorizations of a Coxeter element that have the same length and multiset of conjugacy classes are in the same Hurwitz orbit. This confirms one case of a conjecture of Lewis and Reiner.
The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m, p, n)$ of complex ref…
We consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any permutation into 3-cycles has been worked out by David, Goulden and Jackson. Also, such factorizations for transpositions, the case d = 2, have been considered by Crescimanno and Taylor. Goulden and Jackson have proved the differential equation for the generating series of simple Hurwitz numbers. Based on their results, we use W-operator to prove a differenti…
For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.
In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.
Free Products and the Isomorphism between Standard and Dual Artin Groups
本文研究标准 Artin 群与对偶 Artin 群之间满同态 $Ψ$ 为同构的条件，利用非交叉划分偏序集上的 Hurwitz 作用传递性和稳定化子条件给出判定，并证明这些条件在自由积和直积下保持。
A duality for labeled graphs and factorizations with applications to graph embeddings and Hurwitz enumeration
本文为置换因子分解与边、顶点均带标记的图建立心身对偶，借助恰当嵌入图解释并揭示其与辫群Hurwitz作用及胞腔嵌入图理论的联系，为图嵌入与Hurwitz计数提供理论基础。
In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a $d$-cycle into a product of cycles of prescribed lengths is $d^{r-2}.$ To prove our result, we first define a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the cardinality of this new class which with proper parameters is exactly $d^{r-2}.$ The main part of this paper is the proof that there is a bijection from factorizations of a $d$-cycle to multi-noded rooted trees via factorization graphs. This implies the d…
We consider a generalization of the fundamental theorem of finitely generated abelian groups for some non-abelian groups, which is called OGS. First, we consider the dihedral group, which is a non-abelian extension of an abelian group by an involution. Then, we focus on a special case, where the abelian group is cyclic, which is the two-generated Coxeter group I{2}(m). We mention interesting connections between the reduced Coxeter presentation and a particular OGS canonical presentation, which we call the standard OGS canonical presentation. These connections motivate us to offer a generalizat…
The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relati…
Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element
本研究利用平凡化定理规定的LL映射变体，拓展Bessis等人的工作，对良生成复反射群的Coxeter元的本原因子分解进行计数，深化非交叉格的精细计数与结构性质研究。
Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case
本文研究扩张 Weyl 群与加权射影线上厚子范畴的组合对应，证明管状类型下 Hurwitz 作用在 Coxeter 元约化反射分解上的传递性，从而建立保序双射。
Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups
该研究针对良生成反射群，基于W-拉普拉斯算子谱给出Coxeter元反射分解的加权计数通用公式，推广矩阵树/森林定理，引入Jucys-Murphy元变体与塔等价概念，并拓展至Coxeter组合学其他问题。
We consider three notions of connectivity and their interactions in partially ordered sets coming from reduced factorizations of an element in a generated group. While one form of connectivity essentially reflects the connectivity of the poset diagram, the other two are a bit more involved: Hurwitz-connectivity has its origins in algebraic geometry, and shellability in topology. We propose a framework to study these connectivity properties in a uniform way. Our main tool is a certain linear order of the generators that is compatible with the chosen element.
We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxet…
Coxeter groups and Billey-Postnikov decompositions
本文综述了Billey-Postnikov分解在光滑Schubert簇分类计数、反演超平面排列及排列模式避免等领域的应用。
Twisted Hurwitz numbers: Tropical and polynomial structures
本文针对 b=1 时的扭曲 Hurwitz 数，建立其基于热带覆盖的热带解释，并研究其多项式结构。
Factorization of Temperley--Lieb diagrams
本文针对 Temperley--Lieb 代数中的图，提出一种高效算法，能从任意图重构其对应的约化分解，解决了从图到分解的逆向难题。
We describe an approach, via Malle's permutation $Ψ$ on the set of irreducible characters $\text{Irr}(W)$, that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements.
Continuous Noncrossing Partitions and Weighted Circular Factorizations
本文研究了复平面单位圆上的连续非交叉划分，证明其构成拓扑偏序集并同胚于辫群的对偶Garside分类空间，并将此推广至任意Coxeter群中基于Coxeter元的加权线性分解构造。
We study a family of infinite-type Coxeter groups defined by the avoidance of certain rank 3 parabolic subgroups. For this family, rationally smooth elements can be detected by looking at only a few coefficients of the Poincaré polynomial. We also prove a factorization theorem for the Poincaré polynomial of rationally smooth elements. As an application, we show that a large class of infinite-type Coxeter groups have only finitely many rationally smooth elements. Explicit enumerations and descriptions of these elements are given in special cases.
Bargain hunting in a Coxeter group
本文推广Petersen和Tenner的深度统计量至经典Weyl群，定义反射代价为组合表示中整数距离，并给出计算任意元素代价的简单内在公式。
Factorization problems in complex reflection groups
本文枚举良生成复反射群中Coxeter元素的任意因子分解并追踪固定空间维数，给出组合方法及对$W$-非交叉划分偏序的新刻画。
We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic formulation of Q-systems of twisted type. It also yields a uniform proof of the discrete integrability of these Q-systems by identifying them with the dynamics of factorization mappings on quotients of double Bruhat cells. On the double Bruhat cell itself, we find these dynamics are closely related to those of the Fomin-Zelevinsky twist map. This leads to an exp…
Reflection groups, geometry of the discriminant and noncrossing partitions. When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. This structure appears in several algebraic setups (dual braid monoid, cluster algebras...). Many combinatorial properties of NCP_W are proved case-by-case, using the classification of reflection groups. It is the case for Chapoton's formula, expressing the number of multichains of a given length in the lattice NCP_W, in t…
The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a chara…
Counting factorizations of Singer cycles in linear and unitary groups
本文计数有限域上酉群与线性群中 Singer 循环的反射因式分解，得到最小长度情形下的乘积公式，并揭示 Ennola 对偶现象，方法基于 Deligne--Lusztig 特征标理论。
Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,n}, are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of S_n, and show that they have a polynomiality property. These centrality and polynomiality properti…
For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by the average of length and reflection length. In this article, we focus on the case of the symmetric group, where we show that depth is equal to sum_i max{w(i)-i, 0}. We characterize those permutations for which depth equals length: these are the 321-avoiding permutations (and hence are enumerated by the Catalan numbers). We also characterize those permutations for which depth equals reflection le…
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