Ben Fairbairn, Jürgen Müller
We provide involutory symmetric generating sets of finitely generated Coxeter groups, fulfilling a suitable finiteness condition, which in particular is fulfilled in the finite, affine and compact hyperbolic cases.
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In this article, we first show that in case $n$ is even which Coxeter element in $\mathfrak{S}_{n}$ affords the longest by taking its power to $n/2$. We also show that in case $n$ is odd which Coxeter element affords the longest in $\mathfrak{S}_{n}$.
We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.
The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.
The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. In this paper we complete the calculation of the strong symmetric genus for each finite Coxeter group excluding the group E8.
The OGS for non-abelian groups is an interesting generalization of the basis of finite abelian groups. The definition of OGS states that every element of a group has a unique presentation as a product of some powers of specific generators of the group, in a specific given order. In case of the symmetric groups Sn there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group, which is called the standard OGS of Sn. In this paper we generalize the standard OGS of Sn to the finite classical Coxeter group Bn. We …
The OGS for non-abelian groups is an interesting generalization of the basis of finite abelian groups. The definition of OGS states that every element of a group has a unique presentation as a product of some powers of specific generators of the group, in a specific given order. In case of the symmetric groups Sn there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group, which is called the standard OGS of Sn. In this paper we generalize the standard OGS of Sn to the finite classical Coxeter group Dn. We …
We study Coxeter groups from which there is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set $T$ of transpositions. These quotients, denoted here by C_Y(T), are a special type of the generalized Coxeter groups defined in \cite{CST}, and also arise in the computation of certain invariants of surfaces. We use a surprising action of $S_n$ on the kernel of the surjection $C_Y(T) \ra S_n$ to show that this kernel embeds in the direct product of $n$ copies of the free group $π_1(T)$ (with the exception …
Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite ar…
Finite Coxeter Groups and Generalized Elnitsky Tilings
本文基于Elnitsky对A、B、D型Coxeter群约化词与多边形铺砌的双射，通过嵌入对称群的方法，将其推广至所有有限Coxeter群。
Given an odd prime number $p$ and a Coxeter group $W$ such that the order of the product $st$ is prime to $p$ for every Coxeter generators $s,t$ of $W$, we prove that the $p$-local homology groups $H_k(W,\mathbb{Z}_{(p)})$ vanish for $1\leq k\leq 2(p-2)$. This generalize a known vanishing result for symmetric groups due to Minoru Nakaoka.
Cactus groups are traditionally defined based on symmetric groups, and pure cactus groups are particular subgroups of cactus groups. Mostovoy showed that pure cactus groups embed into right-angled Coxeter groups. We generalize this result to cactus groups associated with arbitrary finite Coxeter groups and we investigate some representations of generalized cactus groups and deduce the linearity of generalized cactus groups.
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。
There were defined by R. Shwartz OGS for non-abelian groups, as an interesting generalization of the basis of finite abelian groups. The definition of OGS states that that every element of a group has a unique presentation as a product of some powers of the OGS, in a specific given order. In case of the symmetric groups S_{n} there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group. The OGS presentation helps us to find the Coxeter length and the descent set of an arbitrary element of the symmetric group…
Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products
本文通过Goursat引理将有限Coxeter群中抛物型子群的正规化子视为子直积，并利用格的Galois连接提出新的分解方法，为研究提供了新的对称视角和分类结果。
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.
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。
We say a permutation $π=π_1π_2\cdotsπ_n$ in the symmetric group $\mathfrak{S}_n$ has a peak at index $i$ if $π_{i-1}<π_i>π_{i+1}$ and we let $P(π)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$ is a peak of $π$}\}$. Given a set $S$ of positive integers, we let $P (S; n)$ denote the subset of $\mathfrak{S}_n$ consisting of all permutations $π$, where $P(π) =S$. In 2013, Billey, Burdzy, and Sagan proved $|P(S;n)| = p(n)2^{n-\lvert S\rvert-1}$, where $p(n)$ is a polynomial of degree $\max(S)- 1$. In 2014, Castro-Velez et al. considered the Coxeter group of type $B_n$ as the group of signed per…
In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of $W$, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.
We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.
We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities…
Let $W^c(A_n)$ be the set of fully commutative elements of the Coxeter group $W(A_n)$. Let $$ a_n(q)= \sum_{w \in W^c(A_n)} q^{l(w)} . $$ We compute $a_n(q)$.
Group actions on complexes, Kozsul models, presentations, and a theorem of Coxeter
本文基于Kozsul思想重新证明了Brown关于离散群在单连通复形上作用可构造模顶点稳定子展示的定理，并通过基本群、对称群及二十面体相关群等例子展示其简化过程。
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to a component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in…
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras: a novel 2-parameter type $A$ family that we call $NC_A(n,d)$. We explore several combinatorial properties of $NC_A(n,d)$, including its Coxeter word basis, length function, and Hilbert-Poincare series, and show that the corresponding generalized Coxeter group is not a flat deformation of $NC_A(n,d)$. These algebras yield symmetric semigroup module categorie…
We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no a…
Let $Σ$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $Σ$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $Γ$, and an infinite family of uniform lattices with covolumes converging to that of $Γ$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $Γ$ is not finitely generated. Examples of $Σ$ to which ou…
We refine a conjecture by Lehrer and Solomon on the structure of the Orlik-Solomon algebra of a finite Coxeter group $W$ and relate it to the descent algebra of $W$. As a result, we claim that both the group algebra of $W$, as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of $W$. We give a uniform proof of the claim for symmetric groups. In addition, we prove that a relative version of the conjecture holds for every pair $(W, W_L)$, where $W$ is arbitrary and $W_…
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…
In recent papers we have refined a conjecture of Lehrer and Solomon expressing the characters of a finite Coxeter group $W$ afforded by the homogeneous components of its Orlik-Solomon algebra as sums of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture also relates the Orlik-Solomon characters above to the terms of a decomposition of the regular character of $W$ related to the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of the Orlik-Solomon algebra have parallel, graded de…
For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).
This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to relate…
The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule a…
Congruence subgroups and crystallographic quotients of small Coxeter groups
本文研究小 Coxeter 群的全等子群性质，证明无限且非 virtually abelian 的小 Coxeter 群不满足该性质，并确定孪生群与三重群的晶体学商群及其维数和 holonomy 表示。
We introduce a new statistic on the hyperoctahedral groups (Coxeter groups of type B), and give a conjectural formula for its signed distributions over arbitrary descent classes. The statistic is analogous to the classical Coxeter length function, and features a parity condition. For descent classes which are singletons the conjectured formula gives the Poincaré polynomials of the varieties of symmetric matrices of fixed rank. For several descent classes we prove the conjectural formula. For this we construct suitable "supporting sets" for the relevant generating functions. We prove cancellati…
For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm{AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb{H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and find examples of Coxeter groups $W$ such that the s…
Stratifying the space of barcodes using Coxeter complexes
本研究运用几何群论工具，基于对称群抛物子群的带标记双陪集对含n条条带的条形码空间分层，得到推广现有不变量的新条形码不变量，并衍生出与修正瓶颈、Wasserstein度量一致的空间度量。
On Codes in (Generalized) Symmetric Groups
本文证明对称群S_n中长度至少为3的Young子群不是关于共轭类的码，并刻画了某些子群成为码的条件，同时描述了B_n、C_n、D_n型Coxeter群及广义对称群中的码。
Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity
本文利用双模系数的 Schatten 类条件刻画弱包含关系，并研究 Coxeter 群梯度双模，建立 gradient-S_p、紧化小性与奇偶路径的等价性，从而推广强刚性结果。
There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
Commutators of signed $n$-cycles
本文证明在 $n\geq6$ 时，对称群和带符号对称群的换位子群元素均可表示为两个 $n$-循环的换位子，仅当 $n\equiv2\pmod4$ 时 $-\mathrm{id}$ 例外，并给出 Coxeter 群类型 $A$、$B$ 的相应描述。
We investigate a certain linear combination $K(\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\{(a,b,c,d,e,f,g)\in\Bbb C^7\colon e+f+g-a-b-c-d=1\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $μ_1,μ_2,μ_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and cont…
In theory of Coxeter groups, bigrassmannian elements are well known as elements which have precisely one left descent and precisely one right descent. In this article, we prove formulas on enumeration of bigrassmannian permutations weakly below a permutation in Bruhat order in the symmetric groups. For the proof, we use equivalent characterizations of bigrassmannian permutations by Lascoux-Schutzenberger and Reading.
Absolute moved spaces and noncrossing partition posets in arbitrary Coxeter groups
本文提出绝对移动空间概念，替代任意Coxeter群中非交叉划分偏序集元素的移动空间，保持维数且具单射性，并证明反射子群映射单射，给出秩三格性质新证明及秩四非格例子。
Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some integer power of each generator. These include usual nil-Coxeter algebras, nil-Temperley-Lieb algebras, and their variants, and lead to symmetric semigroup module categories which necessarily cannot be monoidal. Motivated by classical work of Coxeter (1957) and the Broue-Malle-Rouquier freeness conjecture [Crelle 1998], and continuing beyond work of the sec…
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