Group Theory

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Showing new listings for Friday, 26 September 2025

New submissions (showing 2 of 2 entries )

[1] arXiv:2509.20532 [cn-pdf, pdf, html, other]

Title: A notion of quasi-convex subgroups in acylindrically hyperbolic groups Show Chinese title

Title: 在无矛盾双曲群中的拟凸子群的概念

Ping Wan

Subjects: Group Theory (math.GR)

In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off cusped spaces. We also prove that this notion of quasiconvexity is preserved under sufficiently long Dehn filling. As an application, we generalize a theorem of Groves-Manning on groups acting on $\operatorname{CAT}(0)$ cube complexes. Show Chinese abstract

在本文中，我们提出了一个在具有双曲嵌入子群的有限生成群设定下的拟凸性概念。 我们的主要结果表明，这种概念在锥化尖点空间的设定下能给出一致的拟凸常数。 我们还证明，这种拟凸性在足够长的Dehn填充下是保持的。 作为应用，我们推广了Groves-Manning关于作用在$\operatorname{CAT}(0)$个立方复形上的群的一个定理。

[2] arXiv:2509.21139 [cn-pdf, pdf, other]

Title: Rigid automorphisms of linking systems of finite groups of Lie type Show Chinese title

Title: 有限李型群的链接系统的刚性自同构

Jonathon Villareal

Subjects: Group Theory (math.GR)

Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$. An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system. An inner rigid automorphism is conjugation by some element of the center of $S$. If $\mathcal{L}$ is the centric linking system of a finite group $G$, then rigid automorphisms of $\mathcal{L}$ are closely related to automorphisms of $G$ that centralize $S$. For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2. We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism. Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic. These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant. Show Chinese abstract

设 $\mathcal{L}$是与有限 $p$-群 $S$上的饱和融合系统相关联的中心连接系统。 $\mathcal{L}$的自同构被称为刚性，如果它在融合系统上限制为恒等映射。 内刚性自同构是通过 $S$的中心中的某个元素共轭得到的。 如果$\mathcal{L}$是有限群$G$的中心链接系统，那么$\mathcal{L}$的刚性自同构与中心化$S$的$G$的自同构密切相关。 对于奇素数，所有的刚性自同构都是内自同构，但对于素数 2 这并不成立。 我们确定在素数 2 下哪些已知的拟单链接系统具有非内刚性自同构。 基于之前的结果，这简化为处理素数 2 下奇特征的单李型有限群的链接系统的情况。 这些群没有非内刚性自同构，但有两个例外族：二维射影特殊线性群以及二次型非平方判别式的偶维正交群。

Cross submissions (showing 2 of 2 entries )

[3] arXiv:2509.20929 (cross-list from math-ph) [cn-pdf, pdf, html, other]

Title: Complex Lies, Real Physics: The Role of Algebra Complexification Show Chinese title

Title: 复数李，实物理：代数复化的作用

Tanguy Marsault (CEA Saclay), Laurent Schoeffel (CEA Saclay)

Comments: 17 pages

Subjects: Mathematical Physics (math-ph) ; High Energy Physics - Theory (hep-th) ; Group Theory (math.GR)

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessary real. In most cases, the complexification of those Lie algebra is necessary in order to derive irreductible representations of the Lie algebra and subsequently of the symmetry group. In this paper, we give a precise definition of the concept and prove step by step an important result $$\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}. $$ This result is used to determine the irreductible representations of the proper Lorentz group and thus the physical objects admissible when this symmetry is present. It is shown that finite representations of the proper Lorentz group are characterized by pairs of half-integers $(j_1,j_2)$, which determine unambiguously the physical object associated to the given representation. For example, the representation $(0,0)$ of dimension $1$ is called the scalar representation, it corresponds to the Higgs field, and $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ of dimension $4$ is called the Dirac spinor representation, it corresponds to matter particle called fermions. This means that the mathematical group structure determines the material content of the universe following this algebraic structure. Show Chinese abstract

在物理学中，李群表示描述给定系统对称变换的代数结构。 然后，这些群的下降李代数是必要的实数。 在大多数情况下，为了推导李代数的不可约表示，进而推导对称群的不可约表示，需要对这些李代数进行复化。 在本文中，我们给出了该概念的精确定义，并逐步证明了一个重要的结果$$\left(\mathfrak{g}^\mathbb{R}\right)_\mathbb{C} \simeq \mathfrak{g} \times \bar{\mathfrak{g}}. $$。这个结果用于确定适当洛伦兹群的不可约表示，从而确定当存在这种对称性时允许的物理对象。 结果显示，适当洛伦兹群的有限表示由一对半整数$(j_1,j_2)$表征，这些半整数明确地决定了与给定表示相关的物理对象。 例如，维度为$1$的表示$(0,0)$称为标量表示，它对应于希格斯场，而维度为$4$的表示$(\frac{1}{2},0) \oplus (0,\frac{1}{2})$称为狄拉克旋量表示，它对应于称为费米子的物质粒子。 这意味着根据这种代数结构，数学群结构决定了宇宙的物质内容。

[4] arXiv:2509.20945 (cross-list from math.AG) [cn-pdf, pdf, html, other]

Title: Subgroups of the Projective Linear Group Realized by wild Galois Points Show Chinese title

Title: 射影线性群的子群由野性伽罗瓦点实现

Taro Hayashi, Kashu Ito, Atsuya Nakajima, Keika Shimahara

Subjects: Algebraic Geometry (math.AG) ; Group Theory (math.GR)

We work over an algebraically closed field of positive characteristic. This paper investigates linear representations of Galois groups arising from wild Galois points on projective hypersurfaces. We prove that these Galois groups lift to the general linear group and act naturally on vector spaces. Furthermore, we establish necessary and sufficient conditions for subgroups of the projective linear group to be realized as Galois groups of wild Galois points. In addition, we show that projections from wild Galois points on normal hypersurfaces are necessarily wildly ramified. We provide a geometric criterion for detecting wild ramification via the fixed loci of birational automorphisms, linking group-theoretic properties to the geometry of the hypersurface. Show Chinese abstract

我们在正特征的代数闭域上进行工作。 本文研究了从射影超曲面上的野性伽罗瓦点产生的伽罗瓦群的线性表示。 我们证明这些伽罗瓦群可以提升到一般线性群，并在向量空间上自然作用。 此外，我们建立了射影线性群的子群作为野性伽罗瓦点的伽罗瓦群的必要且充分条件。 另外，我们证明了从正常超曲面上的野性伽罗瓦点进行的投影一定是野性分歧的。 我们提供了一个几何准则，通过双有理自同构的固定点来检测野性分歧，将群论性质与超曲面的几何联系起来。

Replacement submissions (showing 4 of 4 entries )

[5] arXiv:2206.13275 (replaced) [cn-pdf, pdf, html, other]

Title: Cuts, flows and gradient conditions on harmonic functions Show Chinese title

Title: 割集、流和调和函数的梯度条件

Antoine Gournay

Comments: 23 pages, many sections cut out

Subjects: Group Theory (math.GR) ; Representation Theory (math.RT)

Reduced cohomology motivates to look at harmonic functions which satisfy certain gradient conditions. If $G$ is a direct product of two infinite groups or a (FC-central)-by-cyclic group, then there are no harmonic functions with gradient in $c_0$ on its Cayley graphs. From this, it follows that a metabelian group $G$ has no harmonic functions with gradient in $\ell^p$. Show Chinese abstract

减少上同调促使人们研究满足某些梯度条件的调和函数。 如果$G$是两个无限群的直积，或者是一个 (FC-中心)-循环群，那么在其凯莱图上不存在梯度属于$c_0$的调和函数。 由此可得，一个可解群$G$在其凯莱图上不存在梯度属于$\ell^p$的调和函数。

[6] arXiv:2309.02176 (replaced) [cn-pdf, pdf, html, other]

Title: Kac-Moody Symmetric Spaces: arbitrary symmetrizable complex or almost split real type Show Chinese title

Title: Kac-Moody对称空间：任意可对称化复数或几乎分裂实数类型

Ralf Köhl, Christian Vock

Subjects: Group Theory (math.GR)

Kac-Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and the first-named author for centered Kac-Moody groups, that is, Kac-Moody groups that are generated by their root subgroups. In the case of non-invertible generalized Cartan matrices this leads to complications that -- within the approach proposed originally -- cannot be repaired in the affine case. In the present article we propose an alternative approach to Kac-Moody symmetric spaces which for invertible generalized Cartan matrices provides exactly the same concept, which for the non-affine non-invertible case provides alternative Kac-Moody symmetric spaces, and which finally provides Kac-Moody symmetric spaces for affine Kac-Moody groups. In a nutshell, the original intention by Freyn, Hartnick, Horn and K\"ohl was to construct symmetric spaces that likely lead to primitive actions of the Kac-Moody groups; this, of course, cannot work in the affine case as affine Kac-Moody groups are far from simple. Additionally, we study the Galois descent to almost split real Kac-Moody symmetric spaces based on the theory of almost split Kac-Moody groups developed by R\'emy 2002. Show Chinese abstract

Kac-Moody对称空间由Freyn、Hartnick、Horn和第一作者为中心Kac-Moody群引入，即由其根子群生成的Kac-Moody群。 在广义Cartan矩阵不可逆的情况下，这会导致一些复杂性——在最初提出的方案中，这些复杂性在仿射情况下无法修复。 在本文中，我们提出了一种Kac-Moody对称空间的替代方法，对于可逆广义Cartan矩阵，它提供了完全相同的概念，对于非仿射不可逆情况，它提供了替代的Kac-Moody对称空间，并最终为仿射Kac-Moody群提供了Kac-Moody对称空间。 简而言之，Freyn、Hartnick、Horn和Köh1的原始意图是构造可能导向Kac-Moody群的原始作用的对称空间；当然，在仿射情况下这无法实现，因为仿射Kac-Moody群距离简单群还很远。 此外，我们基于Rémy于2002年发展的几乎分裂Kac-Moody群理论，研究了到几乎分裂实Kac-Moody对称空间的伽罗瓦下降。

[7] arXiv:2504.14646 (replaced) [cn-pdf, pdf, html, other]

Title: Bol loops of order 27 Show Chinese title

Title: 27阶 Bol 代数

Alexander Grishkov, Michael Kinyon, Petr Vojtěchovský

Comments: 12 pages, amsart; v2. changes in response to (excellent) referee report

Subjects: Group Theory (math.GR)

We classify Bol loops of order $27$, using a combination of theoretical results and computer search. There are $15$ Bol loops of order $27$, including five groups. New constructions for the ten nonassociative Bol loops of order $27$ are given. Show Chinese abstract

我们通过理论结果和计算机搜索的结合，对阶为$27$的 Bol 代数进行分类。 阶为$27$的 Bol 代数共有$15$个，其中包括五个群。 给出了阶为$27$的十个非结合 Bol 代数的新构造方法。

[8] arXiv:2503.20735 (replaced) [cn-pdf, pdf, html, other]

Title: Weighted Orlicz $*$-algebras on locally elliptic groups Show Chinese title

Title: 加权Orlicz $*$-代数在局部椭圆群上

Max Carter

Comments: 33 pages. Some minor changes have been made to the exposition. Accepted for publication in Studia Mathematica

Subjects: Functional Analysis (math.FA) ; Group Theory (math.GR) ; Operator Algebras (math.OA) ; Representation Theory (math.RT)

Let $G$ be a locally elliptic group, $(\Phi,\Psi)$ a complementary pair of Young functions, and $\omega: G \rightarrow [1,\infty)$ a weight function on $G$ such that the weighted Orlicz space $L^\Phi(G,\omega)$ is a Banach $*$-algebra when equipped with the convolution product and involution $f^*(x):=\overline{f(x^{-1})}$ ($f \in L^\Phi(G,\omega)$). Such a weight always exists on $G$ and we call it an $L^\Phi$-weight. We assume that $1/\omega \in L^\Psi(G)$ so that $L^\Phi(G,\omega) \subseteq L^1(G)$. This paper studies the spectral theory and primitive ideal structure of $L^\Phi(G,\omega)$. In particular, we focus on studying the Hermitian, Wiener and $*$-regularity properties on this algebra, along with some related questions on spectral synthesis. It is shown that $L^\Phi(G,\omega)$ is always quasi-Hermitian, weakly-Wiener and $*$-regular. Thus, if $L^\Phi(G,\omega)$ is Hermitian, then it is also Wiener. Although, in general, $L^\Phi(G,\omega)$ is not always Hermitian, it is known that Hermitianness of $L^1(G)$ implies Hermitianness of $L^\Phi(G,\omega)$ if $\omega$ is sub-additive. We give numerous examples of locally elliptic groups $G$ for which $L^1(G)$ is Hermitian and sub-additive $L^\Phi$-weights on these groups. In the weighted $L^1$ case, even stronger Hermitianness results are formulated. Show Chinese abstract

设$G$是一个局部椭圆群，$(\Phi,\Psi)$是一个互补的 Young 函数对，$\omega: G \rightarrow [1,\infty)$是在$G$上的一个权函数，使得加权 Orlicz 空间$L^\Phi(G,\omega)$在配备卷积乘积和对合$f^*(x):=\overline{f(x^{-1})}$($f \in L^\Phi(G,\omega)$) 时是一个 Banach$*$-代数。 这样的权在$G$上总是存在，我们称之为$L^\Phi$-权。 我们假设$1/\omega \in L^\Psi(G)$，以便$L^\Phi(G,\omega) \subseteq L^1(G)$。 本文研究$L^\Phi(G,\omega)$的谱理论和原始理想结构。 特别是，我们专注于研究该代数上的厄米特、维纳和$*$-正则性性质，以及与谱合成相关的某些问题。 可以看出，$L^\Phi(G,\omega)$始终是准厄米特的，弱维纳的且$*$正则的。 因此，如果$L^\Phi(G,\omega)$是厄米特的，那么它也是维纳的。 尽管一般情况下，$L^\Phi(G,\omega)$并不总是厄米特的，但已知当$\omega$是次可加的时候，$L^1(G)$的厄米特性意味着$L^\Phi(G,\omega)$的厄米特性。 我们给出许多局部椭圆群$G$的示例，其中$L^1(G)$是这些群上的厄米特且次可加的$L^\Phi$权函数。在加权$L^1$的情况下，给出了更强烈的厄米特性结果。