On double cosets in free groups

关于双陪集可分性与Wilson-Zalesskii性质

On double coset separability and the Wilson-Zalesskii property

本文证明了每个双陪集可分群都享有Wilson-Zalesskii性质，并构造了一个不具备该性质且非双陪集可分的LERF群实例。

A Classification of Certain Finite Double Coset Collections in the Classical Groups

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.

Infiniteness of Double Coset Collections in Algebraic Groups

Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show th…

A note on Bruhat order and double coset representatives

Let $W$ be a finite coxeter group, let $W_I$, $W_J$ be standard parabolic subgroups, let $u$, $v$ be minimal double cosets representatives of double cosets in $W_I W / W_J$ and let $u'$, $v'$ be the maximal representatives of those same two cosets. We show that $u < v$ if and only if $u' < v'$ for the Bruhat order.

通过子因子获取子群陪集代表系和双重陪集代表系的完备算法

Complete algorithms for transversals and double coset representatives of subgroups via subfactors

本文证明了获取子群所有陪集代表系的完备算法，并提出一种通用算法以获取子群所有双重陪集代表系及其数量，同时引入了三个子集的中间直积概念及获取中间因子的方法。

On concentration of convolutions of double cosets at infinite-dimensional limit

For infinite-dimensional groups $G\supset K$ the double cosets $K\setminus G/K$ quite often admit a structure of a semigroup; these semigroups act in $K$-fixed vectors of unitary representations of $G$. We show that such semigroups can be obtained as limits of double cosets hypergroups (or Iwahori--Hecke type algebras) on finite-dimensional (or finite) groups.

On Growth of Double Cosets in Hyperbolic Groups

Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $λ> 0$, such that $gr(H,A,B)(r) >λf_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function…

A double coset formula for the genus of a nilpotent group

We derive double coset formulae for the genus and extended genus of a finitely generated nilpotent group G, using the notions of bounded and bounded above automorphisms of $\prod G_S$, which are defined relative to a fixed fracture square for G.

Construction of double coset system of a Coxeter group and its applications to Bruhat graphs

We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomial…

Finitely presented wreath products and double coset decompositions

We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characte…

Real double coset spaces and their invariants

Let G be a real form of a complex reductive group. Suppose that we are given involutions σand θof G. Let H=G^σdenote the fixed group of σand let K=G^θdenote the fixed group of θ. We are interested in calculating the double coset space H\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles.

双余集zeta函数用于作用于树的群

Double-coset zeta functions for groups acting on trees

本文研究了作用于树的群的双余集zeta函数，通过几何方法和行列式公式揭示了其与欧拉-庞加莱特征及Ihara zeta函数的联系。

String rewriting for Double Coset Systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are m…

On multiplication of double cosets for $\GL(\infty)$ over a finite field

We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural homomorphism $B\to GL(m)$. We show that the space of double cosets of $GL(\infty)$ by $P$ admits a natural structure of a semigroup. In fact this semigroup acts in subspaces of $P$-fixed vectors of some unitary representations of $GL(\infty)$ over finite field.

Double cosets for $\SU(2)\times...\times\SU(2)$ and outer automorphisms of free groups

Consider the space of double cosets of the product of $n$ copies of $\SU(2)$ with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group $F_{n-1}$ and of the braid group of $n-1$ strings.

中心扩张中的乘积可分离性

Product separability in central extensions

本文研究了中心扩张在子群可分离性和双余集可分离性下的性质，证明了在特定条件下中心扩张的乘积可分离性和双余集可分离性的等价性及稳定性。

Infinite symmetric group and combinatorial descriptions of semigroups of double cosets

We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs, two-dimensional surfaces with polygonal tiling) and describe explicitly the multiplications.

Statistical Enumeration of Groups by Double Cosets

Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) $H = K = $ the Borel subgroup in $GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside $S_{2n}$, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if $H$ and $K$ are parabolic subgroups of $S_n$, the double cosets are `contingen…

Infinite tri-symmetric group, multiplication of double cosets, and checker topological field theories

We consider a product of three copies of infinite symmetric group and its representations spherical with respect to the diagonal subgroup. We show that such representations generate functors from a certain category of simplicial two-dimensional surfaces to the category of Hilbert spaces and bounded linear operators.

On double cosets of groups $GL(n)$ with respect to subgroups of block strictly triangular matrices

We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.

计数双陪集及其在一般三维流形中的应用

Counting double cosets with application to generic 3-manifolds

本研究推广了双陪集增长理论至更广群类，证明 Morse 子群双陪集增长与轨道增长相当，并应用于证实 Maher 猜想：双曲三维流形在 Heegaard 分解流形中指数泛型。

Separability of double cosets and conjugacy classes in 3-manifold groups

Let M = H^3 / Γbe a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γand g is in Γ, then the double coset HgK is separable in Γ. As a consequence we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of M is conjugacy separable then so is the fundamental group of M. Invoking recent work of Agol and Wise, it follows that if M is a compact, orientable 3-manifold then π_1(M) is conjugacy separable.

辫群中的Hilden双陪集问题

The Hilden Double Coset Problem in Braid Groups

本文解决了辫群模Hilden子群的双陪集问题，证明平结闭包的链环问题稳定等价于可解代数问题，并利用Garside分解方法。

A note on Bruhat decomposition of GL(n) over local principal ideal rings

Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A^n. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend on the ring in question. For n=3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.

群上马尔可夫链双陪集归并的一个充要条件及其在随机置顶洗牌中的应用

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle

本文给出群上随机游走在双陪集空间上诱导马尔可夫链的充要条件，并应用该结果分析随机置顶洗牌的一个变体，确定其不变分布、特征值及收敛速率。

对称群抛物双陪集中的不动点、下降与逆序

Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

本文研究对称群固定抛物双陪集中随机排列的不动点、下降与逆序统计量，利用 Stein 方法、大小偏倚耦合和依赖图证明 Poisson 极限、中心极限定理、收敛速率及测度集中性。

A proof of the polycirculant conjecture

This paper presents a solution of the polycirculant conjecture which states that every vertex-transitive graph G has an automorphism that permutes the vertices in cycles of the same length. This is done by identifying vertex-transitive graphs as coset graphs. For a coset graph H, an equivalence relation $\sim$ is defined on the vertices of cosets with classes as double cosets of the stabiliser and any other proper subgroup A' of a transitive group A of G. Induced left translations of elements of the subgroup A' are semi-regular since they preserve these double cosets and acts regularly on each…

对称子群的大双余集积

Large products of double cosets for symmetric subgroups

研究分类满足KxKyK=G的x,y对，给出一般简单紧连通李群G的必要条件，并在G=SU(n)时完全分类，排除AIII类型子群。应用于量子计算门分解。

用 Clifford 双陪集界定纠缠熵

Bounding Entanglement Entropy with Clifford Double Cosets

本文提出收缩图与 Clifford 双陪集形式体系，刻画量子态在 Clifford 线路下熵向量的可达结构，并给出任意 n 量子比特线路可生成熵向量数量的上界，为全息对偶邻近性提供新视角。

Modules for algebraic groups with finitely many orbits on totally singular 2-spaces

This is the author's second paper treating the double coset problem for classical groups. Let $G$ be an algebraic group over an algebraically closed field $K$. The double coset problem consists of classifying the pairs $H,J$ of closed connected subgroups of $G$ with finitely many $(H,J)$-double cosets in $G$. The critical setup occurs when one of $H,J$, say $H$, is reductive, and $J$ is a parabolic subgroup. Assume that $G$ is a classical group, $H$ is simple and $J$ is a maximal parabolic $P_k$, the stabilizer of a totally singular $k$-space. Then most candidates have $k=1$ or $k=2$. The case…

$W(E_{10})$中的谱增长：双陪集过滤与希尔伯特几何

Spectral Growth in $W(E_{10})$: Double Coset Filtration and Hilbert Geometry

本文针对双曲考克斯特群$W(E_{10})$，引入基于共轭反射的双陪集过滤与有根DAG分类方法，结合希尔伯特几何建立谱半径归纳计算框架，结论可推广至$n\ge10$的$W(E_n)$族。

Groups $GL(\infty)$ over finite fields and multiplications of double cosets

Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we cons…

Derangements in Cosets of Primitive Permutation Groups

Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

多值群与舒尔环

Multivalued Groups and Schur Rings

该研究关联舒尔环探讨有限对合多值群，证明陪集多值群属舒尔多值群，构造初等阿贝尔群上的交换多值群族并获其渐近界，还证明双陪集群在该族中比例趋于零。

自由群中陪集的递推关系

Recurrence Relations for Cosets in Free Groups

本文针对双生成元自由群$F_2$的子群陪集，推导了陪集内指定既约长度元素计数的递推关系，给出对应常数的计算算法，并证明有限指数含奇长元子群情形下非零常数有限。

对称群关于较小对称群的双陪集代数

Algebra of double cosets of a symmetric group by a smaller symmetric group

针对对称群$S_{α+n}$及其子群$S_n$，研究$S_n$-双不变函数构成的子代数$\boxed{O}[α;n]$，给出其乘法公式（结构常数为Pochhammer符号）和生成元-关系描述，并构造了复参数插值代数族。

关于pro-$p$群中的概率恒等式与陪集恒等式

On probabilistic identities and coset identities in pro-$p$ groups

本文证明了在$σ$-紧$K$-解析群$G$（$K$为非阿基米德局部域）中，概率恒等式即为陪集恒等式，并应用此结果得出紧$K$-解析群及自由构造得到的各种pro-$p$群满足概率Tits交替性，同时通过Lie理论方法研究了几乎自由pro-$p$群和紧$p$-进解析群中的挠概率恒等式。

Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively sm…

关于一族非陪集二值群

On a family of non-coset 2-valued groups

从多项式定义的一类离散时间二值动力系统中提取出二值群作用族，研究其与任意域上的类似结构，证明它们都是双陪集群，并给出非陪集性证明，首次构造了拓扑非陪集非离散拓扑二值群。

有限域上一般线性群的双陪集代数

One algebra of double cosets for a general linear group over a finite field

本文研究有限域上一般线性群关于固定前α个基元素的子群的双陪集卷积代数，用生成元与关系刻画其结构，并证明该代数族可自然插值到任意复数n。

关于对称群相对于 Young 子群的双陪集代数

On algebras of double cosets of symmetric groups with respect to Young subgroups

本文研究对称群中关于 Young 子群的双陪集代数 $Δ$，构造连续参数外推与渐近版本，并建立纯辫群李代数到 $Δ$ 的自然映射，用于拟正则表示重数空间。

Groups equal to a product of three conjugate subgroups

Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formula…

On expansions of non-abelian free groups by cosets of a finite index subgroup

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarskiś problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the $L_Q$-expansion of $F$ in which every $P_q$ is interpreted as the preimage of $q$ in $F$. In particular we prove an analogous result to Sela's generalization of Merzlyakov…

寡态群、部分双射范畴与有限域上超齐次三次空间

Oligomorphic groups, categories of partial bijections, and ultrahomogeneous cubic spaces over finite fields

本文为某类寡态群构造了 Ismagilov–Olshanski 意义下的双陪集乘法，并将约化双陪集范畴实现为部分双射范畴，进而以有限域上带三次型的超齐次空间为例描述其开子群并分类其酉表示。

The space of coset partitions of $F_n$ and Herzog-Schönheim conjecture

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank. We define $Y$ the space of coset partitions of $F_n$ and show $Y$ is a metric space with interesting properties. In a previous paper, we gave some sufficient conditions on the coset partition of $F_n$ that ensure the conjecture is satisfied. Here, we show that …

Regular sets and counting in free groups

In this paper we study asymptotic behavior of regular subsets in a free group F of finite rank, compare their sizes at infinity, and develop techniques to compute the probabilities of sets relative to distributions on F that come naturally from no-return random walks on the Cayley graph of F. We apply these techniques to study cosets, double cosets, and Schreier representatives of finitely generated subgroups of F and also to analyze relative sizes of regular prefixed-closed subsets in F.

On short expressions for cosets of permutation subgroups

Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Maróti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the …

具有上下文无关原像的群的子集

Subsets of groups with context-free preimages

本文研究有限生成群中可识别上下文无关子集，推广Muller-Schupp定理，证明有限可识别上下文无关子集的存在性、所有共轭类可识别上下文无关性以及陪集可识别上下文无关性均与群的几乎自由性或Schreier陪集图拟等距于树等价。

One Variable Equations in Torsion-Free Hyperbolic Groups

Let $Γ$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Γ$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Γ$ is free.

有限群左陪集上左不变随机游走的弱归并

Weak lumping of left-invariant random walks on left cosets of finite groups

本文刻画了有限群左陪集上左不变随机游走弱归并的初始分布与权重条件，给出阿贝尔子群情形下的线性方程组判据，并应用于洗牌模型中顶牌序列的马尔可夫性验证。

基于Post陪集定理的泛幻排列的N元群

N-ary groups of panmagic permutations from the Post coset theorem

研究泛幻排列（模n皇后问题的解）的代数结构，通过Post陪集定理证明其构成二面体子群陪集的N元群推广，并揭示与数论和组合学的联系。