Journal of Algebra最新文献

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Local Hilbert–Schmidt stability 局部希尔伯特-施密特稳定性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-10 DOI: 10.1016/j.jalgebra.2024.08.042

Francesco Fournier-Facio , Maria Gerasimova , Pieter Spaas

We introduce a notion of local Hilbert–Schmidt stability, motivated by the recent definition by Bradford of local permutation stability, and give examples of (non-residually finite) groups that are locally Hilbert–Schmidt stable but not Hilbert–Schmidt stable. For amenable groups, we provide a criterion for local Hilbert–Schmidt stability in terms of group characters, by analogy with the character criterion of Hadwin and Shulman for Hilbert–Schmidt stable amenable groups. Furthermore, we study the (very) flexible analogues of local Hilbert–Schmidt stability, and we prove several results analogous to the classical setting. Finally, we prove that infinite sofic, respectively hyperlinear, property (T) groups are never locally permutation stable, respectively locally Hilbert–Schmidt stable. This strengthens the result of Becker and Lubotzky for classical stability, and answers a question of Lubotzky.

我们引入了局部希尔伯特-施密特稳定性的概念，其动机来自布拉德福德最近对局部置换稳定性的定义，并给出了局部希尔伯特-施密特稳定但非希尔伯特-施密特稳定的（非永久有限）群的例子。通过类比哈德温和舒尔曼对希尔伯特-施密特稳定可配位群的特征判据，我们用群特征为可配位群提供了一个局部希尔伯特-施密特稳定性判据。此外，我们还研究了局部希尔伯特-施密特稳定性的（非常）灵活的类比，并证明了几个与经典情形类似的结果。最后，我们证明了无限sofic（分别是超线性）性质（T）群从来不是局部置换稳定的，分别是局部希尔伯特-施密特稳定的。这加强了贝克尔和卢博茨基关于经典稳定性的结果，并回答了卢博茨基的一个问题。

引用次数: 0

Ranks of quadratic twists of Jacobians of generalized Mordell curves 广义莫德尔曲线雅各布的二次扭曲秩

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-10 DOI: 10.1016/j.jalgebra.2024.08.041

Tomasz Jędrzejak

Consider a two-parameter family of hyperelliptic curves

C_{q, b} : y^{2} = x^{q} - b^{q}

defined over

Q

, and their Jacobians

J_{q, b}

where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve

C_{q, b}

is a quadratic twist by b of

C_{q, 1}

(a generalized Mordell curve of degree q). First, we obtain a few upper bounds for the ranks e.g., if the class number of

Q (ζ_{q})

is odd,

q \equiv 1 (\mod 4)

and any prime divisor of

2 b

not equal to q is a primitive root modulo q then

rank J_{q, b} (Q) \leq (q - 1) / 2

. Then we focus on

q = 5

and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many b with any number of prime factors such that

rank J_{5, b} (Q) = 0

. We deduce as conclusions the complete list (or the bounds for the number) of rational points on

C_{5, b}

in such cases. Finally, we found for any given q infinitely many non-isomorphic curves

C_{q, b}

such that

rank J_{q, b} (Q) \geq 1

考虑定义在 Q 上的双参数超椭圆曲线族 Cq,b:y2=xq-bq，以及它们的雅各布数 Jq,b，其中 q 是奇素数，在不失一般性的前提下，b 是非零的无平方整数。曲线 Cq,b 是 Cq,1 的 b 二次扭曲（q 度的广义莫德尔曲线）。首先，我们会得到一些秩的上限，例如，如果 Q(ζq) 的类数是奇数，q≡1(mod4)，并且 2b 中任何不等于 q 的素除都是 modulo q 的基元根，那么 rankJq,b(Q)≤(q-1)/2。然后，我们把重点放在 q=5 上，得到了可能的最佳约束（1），甚至是秩的精确值（0）。在这种情况下，我们推导出了 C5,b 上有理点的完整列表（或数量的边界）。最后，我们发现对于任意给定的 q，有无限多条非同构曲线 Cq,b，使得 rankJq,b(Q)≥1。

引用次数: 0

Depth zero supercuspidal representations of classical groups into L-packets: The typically almost symmetric case 经典群的零深度超凸表示成 L 包：典型的几乎对称情况

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-10 DOI: 10.1016/j.jalgebra.2024.09.024

Geo Kam-Fai Tam

We classify what we call “typically almost symmetric” depth zero supercuspidal representations of a classical group over a local field of odd residual characteristic into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens [24] in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values. With the same method, we also compute the parity of a (conjugate-)self-dual depth zero supercuspidal representation of a general linear group.

我们把在奇数残差特征局部域上的经典群的 "典型几乎对称 "深度为零的超pidal 表示归类为 L 包。在这种情况下，我们的主要结果解决了 Lust-Stevens [24] 论文中的一个模糊之处，即他们只能把这些表示归入两个或四个 L 包，如果不是一个的话。通过假设 L-packets 中超pidal 表征的预期数量，我们只利用表征的简单性质来证明主要结果。特别是，我们不需要对字符值进行任何深入计算。用同样的方法，我们还计算了一般线性群的（共轭）自双深度为零的超pidal 表示的奇偶性。

引用次数: 0

Realization of Lie superalgebras G(3) and F(4) as symmetries of supergeometries 实现作为超几何对称的列超拉 G(3) 和 F(4)

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-10 DOI: 10.1016/j.jalgebra.2024.08.035

Boris Kruglikov, Andreu Llabrés

For every parabolic subgroup P of a Lie supergroup G the homogeneous superspace

G / P

carries a G-invariant supergeometry. We address the problem whether

g = Lie (G)

is the maximal (local and global) symmetry of this supergeometry in the case of exceptional Lie superalgebras

G (3)

and

F (4)

. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 special cases), thus realizing

G (3)

and

F (4)

as symmetries of supergeometries. This gives 19 inequivalent

G (3)

-supergeometries and 55 inequivalent

F (4)

-supergeometries, in majority of cases (17 resp 52 cases) those being encoded as vector superdistributions. We describe those supergeometries and realize supersymmetry explicitly.

对于 Lie 超群 G 的每一个抛物线子群 P，均相超空间 G/P 都携带一个 G 不变的超几何。我们要解决的问题是，在例外李超群 G(3) 和 F(4) 的情况下，g=Lie(G) 是否是这个超几何的最大（局部和全局）对称性。我们的方法是考虑每种抛物面选择的负梯度Lie超代数，并计算田中-韦斯费勒延长，必要时还原结构群（2 resp 3 特例），从而实现 G(3) 和 F(4) 作为超几何的对称性。这样就得到了 19 个不等价的 G(3) 超几何和 55 个不等价的 F(4) 超几何，其中大多数情况下（17 个和 52 个）这些超几何都是以向量超分布的形式编码的。我们描述了这些超几何，并明确实现了超对称。

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引用次数: 0

On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties 论紧凑黎曼曲面上最大的纯非自由共形作用及其渐近特性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-10 DOI: 10.1016/j.jalgebra.2024.09.012

C. Bagiński , G. Gromadzki , R.A. Hidalgo

A continuous action of a finite group G on a closed orientable surface X is said to be gpnf (Gilman purely non-free) if every element of G has a fixed point on X. We prove that the biggest order

μ (g)

, of a gpnf-action on a surface of even genus

g \geq 2

, is bounded below by 8g and that this bound is sharp for infinitely many even g as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound

8 g + 8

for arbitrary finite continuous actions. We also describe the asymptotic behavior of μ. We define

M

as the set of values of the function

\tilde{μ} (g) = μ (g) / (g + 1)

and its subsets

M_{+}

and

M_{-}

corresponding to even and odd genera g. We show that the set

M_{+}^{d}

, of accumulation points of

M_{+}

, consists of a single number 8. If g is odd, then we prove that

4 g \leq μ (g) < 8 g

. We conjecture that this lower bound is sharp for infinitely many odd g. Finally, we prove that this conjecture implies that 4 is the only element of

M_{-}^{d}

, leading to

M^{d} = {4, 8}

如果 G 的每个元素在 X 上都有一个定点，那么有限群 G 在封闭可定向曲面 X 上的连续作用被称为 gpnf（吉尔曼纯非自由）作用。我们证明，偶数属 g≥2 的曲面上的 gpnf 作用的最大阶 μ(g)，其下限为 8g，并且这个下限对于无穷多个偶数属 g 也是尖锐的。这就为偶数属提供了著名的任意有限连续作用的阿克拉-麦克拉克伦界 8g+8 的 gpnf 作用类似物。我们还描述了 μ 的渐近行为。我们将 M 定义为函数 μ˜(g)=μ(g)/(g+1)的值集及其对应于偶数和奇数属 g 的子集 M+ 和 M-。如果 g 是奇数，那么我们证明 4g≤μ(g)<8g。最后，我们证明这一猜想意味着 4 是 M-d 的唯一元素，从而得出 Md={4,8}。

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引用次数: 0

The characteristic subgroups of an abelian p-group 无性 p 群的特征子群

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-09 DOI: 10.1016/j.jalgebra.2024.10.001

K. Robin McLean

We describe the characteristic subgroups of a transitive abelian 2-group, thus completing Kaplansky's study in section 18 of his classic monograph Infinite abelian groups. This appears to be a new result even for finite abelian 2-groups.

我们描述了反式无边 2 群的特征子群，从而完成了卡普兰斯基在其经典专著《无限无边群》第 18 节中的研究。即使对于有限无边 2 群，这似乎也是一个新结果。

引用次数: 0

Fusion categories over non-algebraically closed fields 非代数闭域上的融合范畴

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-05 DOI: 10.1016/j.jalgebra.2024.09.010

Sean Sanford

Several complications arise when attempting to work with fusion categories over arbitrary fields. Here we describe some of the new phenomena that occur when the field is not algebraically closed, and we adapt tools such as the Frobenius-Perron dimension in order to accommodate these new effects.

在尝试处理任意域上的融合范畴时，会出现一些复杂问题。在这里，我们描述了当场不代数封闭时出现的一些新现象，并调整了弗罗贝纽斯-佩伦维度等工具，以适应这些新影响。

引用次数: 0

Equivariant oriented homology of the affine Grassmannian 仿射格拉斯曼的等变定向同调

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-02 DOI: 10.1016/j.jalgebra.2024.09.009

Changlong Zhong

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the Goresky-Kottwitz-MacPherson (GKM) condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the formal affine Demazure algebra.

我们将仿射格拉斯曼的小副边等变 K-同调性质推广到莱文和莫雷尔意义上的一般定向（共）同调理论。我们使用的主要工具是与仿射根系统相关联的形式仿射 Demazure 代数。更准确地说，我们证明仿射格拉斯曼的小副等变定向同调满足戈尔斯基-科特维茨-麦克弗森（GKM）条件。我们还证明了它的对偶，即小副边等变同调，与形式仿射 Demazure 代数中点的等变定向同调的中心化同构。

引用次数: 0

Lifting elementary Abelian covers of curves 提升曲线的基本阿贝尔盖

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-26 DOI: 10.1016/j.jalgebra.2024.09.007

Jianing Yang

Given a Galois cover of curves f over a field of characteristic p, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is f. In this paper, we consider the case where the Galois groups are elementary abelian p-groups. We prove a combinatorial criterion for lifting an elementary abelian p-cover, dependent on the branch loci of lifts of its p-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, for

p = 2

, we analyze lifts for several families of

{(Z / 2)}^{3}

-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.

给定特性 p 域上曲线 f 的伽罗瓦盖，提升问题问是否存在一个完整混合特性离散估值环上的伽罗瓦盖，其还原为 f。我们证明了一个提升基本无常 p 盖的组合准则，它取决于其 p 循环子盖的提升支点位置。我们还研究了提升的分支点如何在特殊纤维上凝聚。最后，对于 p=2，我们分析了不同导体类型的 (Z/2)3 覆盖的几个族的提升，既有等距支点几何，也有非等距支点几何。

引用次数: 0

Recognition and constructive membership for purely hyperbolic groups acting on trees 作用于树的纯双曲群的承认和构造成员资格

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-26 DOI: 10.1016/j.jalgebra.2024.09.008

Ari Markowitz

We present an algorithm which takes as input a finite set X of automorphisms of a simplicial tree, and outputs a generating set

X^{'}

〈 X 〉

such that either

〈 X 〉

is purely hyperbolic and

X^{'}

is a free basis of

〈 X 〉

, or

X^{'}

contains a non-trivial elliptic element. As a special case, the algorithm decides whether a finitely generated group acting on a locally finite tree is discrete and free. This algorithm, which is based on Nielsen's reduction method, works by repeatedly applying Nielsen transformations to X to minimise the generators of

X^{'}

with respect to a given pre-well-ordering. We use this algorithm to solve the constructive membership problem for finitely generated purely hyperbolic automorphism groups of trees. We provide a Magma implementation of these algorithms, and report its performance.

我们提出一种算法，输入简单树的有限自变集 X，然后输出〈X〉的生成集 X′，这样，要么〈X〉是纯双曲的，且 X′是〈X〉的自由基，要么 X′包含一个非三维的椭圆元素。作为特例，该算法还能判定作用于局部有限树的有限生成群是否离散和自由。该算法基于尼尔森还原法，通过对 X 重复应用尼尔森变换，使 X′ 的生成数最小化，并与给定的前韦尔排序相关。我们使用这种算法来解决有限生成的纯双曲树自形群的构造成员问题。我们提供了这些算法的 Magma 实现，并报告了其性能。

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引用次数: 0

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Journal of Algebra

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