Alternating subgroups of Coxeter groups

关于Coxeter群嵌入Riordan群的可嵌入性

On embeddability of Coxeter groups into the Riordan group

本文探讨了有限群作为Riordan群子群的线性表示，证明了三次对称群在复数域上无忠实表示，但在特征为三的域上可嵌入Riordan群。

On the Alternating runs polynomial in type B and D Coxeter Groups

Wilf showed that the the alternating runs polynomial $R_n(t)$ counting the number of permutations in the Symmetric group is divisible by $(1+t)^m$ where $m = \lfloor (n-2)/2 \rfloor$. Recently, Bóna gave a group action based proof. Type B and D analogues of Wilf's result are known. In this note, we extend Bóna's proof to prove the type B and D analogue.

Counting nearest faraway flats for Coxeter chambers

In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of $W$ in $[X]$. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double counting argument involving Crapo's beta invariant. As a corollary, this a…

On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof we use Solomon's descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.

On the Hurwitz action in affine Coxeter groups

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.

A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements

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.

Perfect models for finite Coxeter groups

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of $W$-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types $\mathsf{A}_{n}$, $\mathsf{B}_n$, $\mathsf{D}_…

On the Hurwitz action in finite Coxeter groups

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.

关于极大二面体反射子群与广义非交叉分拆

On maximal dihedral reflection subgroups and generalized noncrossing partitions

本文仅基于字的组合学给出了 Dyer 关于任意 Coxeter 群中极大二面体反射子群唯一性结果的新证明，并借此证明了绝对序中长度为3的区间构成格，从而推导出对应的区间群为拟Garside群。

The absolute order of a permutation representation of a Coxeter group

A permutation representation of a Coxeter group $W$ naturally defines an absolute order. This family of partial orders (which includes the absolute order on $W$) is introduced and studied in this paper. Conditions under which the associated rank generating polynomial divides the rank generating polynomial of the absolute order on $W$ are investigated when $W$ is finite. Several examples, including a symmetric group action on perfect matchings, are discussed. As an application, a well-behaved absolute order on the alternating subgroup of $W$ is defined.

Generator Sets for the Alternating Group

Although the alternating group is an index 2 subgroup of the symmetric group, there is no generating set that gives a Coxeter structure on it. Various generating sets were suggested and studied by Bourbaki, Mitsuhashi, Regev-Roichman, Vershik-Vserminov and others. In a recent work of Brenti- Reiner-Roichman it is explained that palindromes in Mitsuhashi's generating set play a role similar to that of re ections in a Coxeter system. We study in detail the length function with respect to the set of palindromes. Results include an explicit combinatorial description, a generating function, and an …

Characteristic polynomials of the weak order on classical and affine Coxeter groups

We find a simple product formula for the characteristic polynomial of the permutations with a fixed descent set under the weak order. As a corollary we obtain a simple product formula for the characteristic polynomial of alternating permutations. We generalize these results to Coxeter groups. We also find a formula for the generating function for the characteristic polynomials of classical Coxeter groups, which is then related to affine Coxeter groups.

Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over Z[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of ``quasiparabolic" subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.

On the group of alternating colored permutations

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product $\mathbb{Z}_r \wr S_n$. We present a 'Coxeter-like' presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ and use this point of view to give another expression for the length function. We also use this covering to lift several known parameters of $\mathbb{Z}_{\frac{r}{2}} \wr S_n$ to the group of alternating colored permutations.

证明：在有限实反射群中，对每个抛物型拟Coxeter元素，其反射分解的Hurwitz作用轨道由两个显然不变量区分

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。

Virtually fibering random right-angled Coxeter groups

We show that the Right-Angled Coxeter group $C=C(G)$ associated to a random graph $G\sim \mathcal{G}(n,p)$ with $\frac{\log n + \log\log n + ω(1)}{n} \leq p < 1- ω(n^{-2})$ virtually algebraically fibers. This means that $C$ has a finite index subgroup $C'$ and a finitely generated normal subgroup $N\subset C'$ such that $C'/N \cong \mathbb{Z}$. We also obtain the corresponding hitting time statements, more precisely, we show that as soon as $G$ has minimum degree at least 2 and as long as it is not the complete graph, then $C(G)$ virtually algebraically fibers. The result builds upon the work…

Coxeter群的固定点陪集中最小长度元素与Bruhat序

Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups

研究Coxeter群中固定点陪集上的Bruhat序，刻画最小长度元素并证明其与序极小性等价。

Parabolic double cosets in Coxeter groups

Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \subseteq S$. Double coset are not uniquely presented by triples $(I,w,J)$. W…

On the Induction of p-Cells

We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see arXiv:1510.01556, arXiv:1901.02323) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a right $p$-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of the $p$-canonical basis.

The parabolic coset structure of Bruhat intervals in Coxeter groups

In this paper, we study the decomposition of Bruhat intervals in a Coxeter group with respect to cosets of a parabolic subgroup. Our main result is that the intersection of a lower Bruhat interval with a parabolic coset contains a unique maximal element. As an application, we give a decomposition formula for the Poincaré polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.

A generalization of Euler numbers to finite Coxeter groups

It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed the value in each case of the classification. We consider here another generalization of Euler numbers for finite Coxeter groups, building on Stanley's result about the number of orbits of maximal chains of set partitions. We present a method to compute these integers and obtain the value in each case of…

Classifying quotients on Coxeter groups by isomorphism in Bruhat order

We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with $(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular, the non-trivial isomorphisms fall into a small number of cases which are highly restricted; all have $W$ finite and $W_J$ a maximal parabolic. This has the immediate application of classifying dominant and antidominant blocks of category $\mathcal O$ for Kac-Moody algebras up to equivalence.

Simply-laced Coxeter groups and groups generated by symplectic transvections

Let W be an arbitrary Coxeter group of simply-laced type (possibly infinite but of finite rank), u,v be any two elements in W, and i be a reduced word (of length m) for the pair (u,v) in the Coxeter group W\times W. We associate to i a subgroup Gamma_i in GL_m(Z) generated by symplectic transvections. We prove among other things that the subgroups corresponding to different reduced words for the same pair (u,v) are conjugate to each other inside GL_m(Z). We also generalize the enumeration result of the first three authors (see AG/9802093) by showing that, under certain assumptions on u and v, …

Signed Alternating-runs enumeration in Classical Weyl Groups

The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show that when enumerated with sign taken into account, one gets a {\it neat formula}. As a consequence, we get a near refinement of a result of Wilf on the exponent of $(1+t)$ when it divides the alternating-runs polynomial in the alternating group $\mathcal{A}_n$. Other applications include a moment-type identity and enumeration of alternating permutations in $\…

A singular Coxeter presentation

We enlarge a Coxeter group into a category, with one object for each finite parabolic subgroup, encoding the combinatorics of double cosets. This category, the singular Coxeter monoid, is connected to the geometry of partial flag varieties. Our main result is a presentation of this category by generators and relations. We also provide a new description of reduced expressions for double cosets. We describe all the braid relations between such reduced expressions, and prove an analogue of Matsumoto's theorem. This gives a proper development of ideas first introduced by Geordie Williamson. In typ…

On the growth of cocompact hyperbolic Coxeter groups

For an arbitrary cocompact hyperbolic Coxeter group G with finite generator set S and complete growth function P(x)/Q(x), we provide a recursion formula for the coefficients of the denominator polynomial Q(x) which allows to determine recursively the Taylor coefficients and the pole behavior of the growth function of G in terms of its Coxeter subgroup structure. We illustrate this in the easy case of compact right-angled hyperbolic n-polytopes. Finally, we provide detailed insight into the case of Coxeter groups with at most 6 generators, acting cocompactly on hyperbolic 4-space, by considerin…

Log-concavity of the Excedance Enumerators in positive elements of Type A and Type B Coxeter Groups

The classical Eulerian Numbers $A_{n,k}$ are known to be log-concave. Let $P_{n,k}$ and $Q_{n,k}$ be the number of even and odd permutations with $k$ excedances. In this paper, we show that $P_{n,k}$ and $Q_{n,k}$ are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014. We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasize the following: though strong synchronisation is stronger than …

无环交替管道梦想的格性质

Lattices properties of acyclic alternating pipe dreams

本文针对A型Coxeter群子词复形的无环面，利用管道梦想表示证明了相关猜想，并给出所有无环面均为强无环的充分条件。

Infinite reduced words and the Tits boundary of a Coxeter group

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point ξ, a natural order-preserving correspondence between infinite reduced words which "point towards" ξ, and elements of the ref…

有限不可约Coxeter群的拟抛物Gelfand模型及其典范Hecke结构

Quasiparabolic Gelfand models for finite irreducible Coxeter groups and their canonical Hecke structures

本文分类了有限不可约Coxeter群中由抛物子群诱导特征构造的拟抛物Gelfand模型，并推导了Gelfand对及典范Hecke结构。

Rationally smooth elements of Coxeter groups and triangle group avoidance

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.

带广义Jucys-Murphy权重的Coxeter因子分解与反射群的矩阵树定理

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。