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显示 2025年08月07日， 星期四 新的列表

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[1] arXiv:2508.04054 [中文pdf, pdf, html, 其他]

标题： 表示的张量幂 of (图) 么半群 显示英文标题

标题： Tensor powers of representations of (diagram) monoids

David He, Daniel Tubbenhauer

评论： 19页，许多图表，欢迎提出意见

主题： 表示理论 (math.RT) ; 范畴论 (math.CT) ; 群论 (math.GR)

我们研究有限独异点表示的张量幂，重点关注其合成长度和不可约直和项数的增长行为。 特别关注诸如Temperley-Lieb、Motzkin和Brauer独异点之类的图示独异点。 对于这些例子，我们计算了具体的资料，包括一些特征表，并分析了它们张量幂分解中的模式。 显示英文摘要

We study tensor powers of representations of finite monoids, focusing on the growth behavior of their composition length and the number of indecomposable summands. Special attention is given to diagram monoids such as the Temperley-Lieb, Motzkin, and Brauer monoids. For these examples, we compute explicit data, including some character tables, and analyze patterns in the decomposition of their tensor powers.

[2] arXiv:2508.04060 [中文pdf, pdf, html, 其他]

标题： 傅里叶变换与实李代数上的内窥镜转移 显示英文标题

标题： Fourier transform and endoscopic transfer on real Lie algebras

Cheng Chen, Zhilin Luo

评论： 任何反馈或意见都非常欢迎

主题： 表示理论 (math.RT) ; 数论 (math.NT)

我们证明了实李代数上的内插转移与傅里叶变换交换，所用方法完全是局部的。 显示英文摘要

We prove that the endoscopic transfer on real Lie algebras commutes with the Fourier transform, using methods that are purely local.

[3] arXiv:2508.04168 [中文pdf, pdf, html, 其他]

标题： 关于多虚拟辫群 $M_kVB_n$ 和多焊接辫群 $M_kWB_n$的表示 显示英文标题

标题： On representations of the multi-virtual braid group $M_kVB_n$ and the multi-welded braid group $M_kWB_n$

Vaibhav Keshari, Mohamad N. Nasser, Madeti Prabhakar

主题： 表示理论 (math.RT) ; 一般拓扑 (math.GN)

本文将多重虚拟辫群$M_kVB_n$的复数齐性$2$-局部表示分类为$\mathrm{GL}_n(\mathbb{C})$对$n\geq3$和$k >1$，表明这样的表示正好分为$2^{k+1}+1$种不同的类型，其中除了三种以外都是不忠实的。 此外，本文研究了多重焊接辫群 $M_kWB_n$ 到 $\mathrm{GL}_n(\mathbb{C})$ 的复数齐次 $2$-局部表示，对于 $n\geq3$ 和 $k >1$，确定了 $3 \cdot 2^{k-1} +1$ 表示。 此外，文章包含了一个非局部表示的构造，即$M_2WB_3$，它扩展了已知的辫群在$3$根弦上的LKB表示，即$B_3$，为一般构造$M_kWB_n$的非局部表示指明了一条路径。 显示英文摘要

This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_kVB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, showing that such representations fall into exactly $2^{k+1}+1$ distinct types, out of which except three all are unfaithful. In addition, this paper investigates complex homogeneous $2$-local representations of the multiple welded braid group $M_kWB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, identifying $3 \cdot 2^{k-1} +1$ representations. Moreover, the article includes a construction of a non-local representation of $M_2WB_3$ that extends the known LKB representation of the braid group on $3$ strands, namely $B_3$, making a path towards constructing non-local representations of $M_kWB_n$ in general.

[4] arXiv:2508.04303 [中文pdf, pdf, html, 其他]

标题： 交错代数和仿射赫克代数对于经典$p$-adic 群的有限中心扩张及其在拟幂群中的应用 显示英文标题

标题： Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups

Volker Heiermann, Chenyan Wu

评论： 51页

主题： 表示理论 (math.RT) ; 数论 (math.NT)

对于一个经典$p$-adic 仿射群的有限中心扩张$\tilde{G}$，我们考虑该群的光滑表示范畴中某个类似 Bernstein 的诱导投射生成元的自同态代数$\tilde{G}$。在 Levi 子群可分解的情况下，我们可以计算这个代数，得到类似于第一位作者之前对经典$p$-adic 群所得的结果，表明这个交织代数是带有参数的仿射 Hecke 代数与一个扭曲的有限群代数的扭曲半直积。我们也讨论了一般情况。接着，我们给出了一个应用，即对$p$-adic 超张量群的真实表示范畴的应用。利用 C. M\oe 关于 Howe 对应的结果，我们证明了这些群的 Bernstein 组件等价于经典群的单奇表示范畴的张量积。这推广了第一位作者之前的结论。它意味着$p$-adic 超张量群的真实表示范畴与相应奇特殊正交群及其纯内形变的光滑表示范畴的直和之间存在范畴等价。 显示英文摘要

For a finite central extension $\tilde{G}$ of a classical $p$-adic reductive group, we consider the endomorphism algebra of some induced projective generator \`a la Bernstein of the category of smooth representations of $\tilde{G}$. In the case where the Levi subgroups decompose, we can compute this algebra to get a result similar to the one previously obtained by the first author for classical $p$-adic groups, showing that this intertwining algebra is a twisted semi-direct product of an affine Hecke algebra with parameters by a twisted finite group algebra. We discuss also the general case. We give then an application to the category of genuine representations of a $p$-adic metaplectic group. Using results of C. M\oe glin relative to the Howe correspondence, we show that the Bernstein components of these groups are equivalent to tensor products of categories of unipotent representations of classical groups. This generalizes a previous result of the first author. It implies an equivalence of categories between the category of genuine representations of the $p$-adic metaplectic group and the direct sums of those of smooth representations of the corresponding odd special orthogonal group and its pure inner form.

交叉提交 (展示 8 之 8 条目 )

[5] arXiv:2508.01886 (交叉列表自 math.QA) [中文pdf, pdf, 其他]

标题： 代数操作符的简明介绍 显示英文标题

标题： A Gentle Introduction to Algebraic Operads

Felicia Ferraioli

评论： 学士论文，萨勒诺大学，于2025年7月28日通过答辩，89页

主题： 量子代数 (math.QA)

本文本基于作者的学士论文，介绍了代数操作符理论，这是一种为现代代数提供统一框架的数学形式化方法。我们展示了结合律、交换律和李代数的基本理论如何可以完全作为三个典型操作符：$\mathsf{As}$、$\mathsf{Com}$和$\mathsf{Lie}$的表示范畴来恢复——这些被称为代数的“三大女神”。 采用演绎和自包含的方法，操作符的概念最初以经典的单对象多范畴形式呈现。随后，提供了其他定义——即部分定义和函子定义。 该框架允许将经典的代数概念，如自由对象和商，扩展到操作符上下文中，从而使操作符能够通过生成元和关系进行正式表示。 本工作的核心结果是对操作符与代数之间对应关系的严格证明。我们建立了操作符$\mathsf{As}$、$\mathsf{Com}$和$\mathsf{Lie}$上的代数与其各自经典对应物之间的范畴同构。 因此，本论文强调了操作符理论如何为代数提供一个更高级的语言，将整个代数理论编码在单一的数学对象中。 这种形式化不仅统一了已知的结构，还为诸如“同伦上的代数”等高级概念奠定了基础，在理论物理和几何等领域有应用。 显示英文摘要

This text, based on the author's Bachelor's thesis, introduces the theory of Algebraic Operads, a mathematical formalism that provides a unifying framework for modern algebra. We demonstrate how the fundamental theories of associative, commutative, and Lie algebras can be fully recovered as categories of representations of three archetypal operads: $\mathsf{As}$, $\mathsf{Com}$ and $\mathsf{Lie}$ -- the so-called 'three graces' of algebra. Following a deductive and self-contained approach, the notion of an operad is initially presented in its classical form, as a single-object multicategory. Subsequently, alternative definitions -- namely, the partial and functorial definitions -- are provided. This framework allows for the extension of classical algebraic notions, such as free objects and quotients, to the operadic context, thereby enabling operads to be formally presented through generators and relations. The central result of this work is the rigorous proof of the correspondence between operads and algebras. We establish isomorphisms of categories between the algebras over the operads $\mathsf{As}$, $\mathsf{Com}$ and $\mathsf{Lie}$, and their respective classical counterparts. This thesis thus highlights how the theory of operads offers a higher-level language for algebra, encoding entire algebraic theories within single mathematical objects. This formalism not only unifies known structures but also lays the foundation for advanced concepts, such as 'algebras up to homotopy,' with applications in fields like theoretical physics and geometry.

[6] arXiv:2508.03815 (交叉列表自 hep-th) [中文pdf, pdf, html, 其他]

标题： 从张量代数到双曲Kac-Moody代数 显示英文标题

标题： From Tensor Algebras to Hyperbolic Kac-Moody Algebras

Axel Kleinschmidt, Hannes Malcha, Hermann Nicolai

评论： 33页

主题： 高能物理 - 理论 (hep-th) ; 量子代数 (math.QA) ; 表示理论 (math.RT)

我们提出一种新方法来研究双曲Kac-Moody代数，更具体地说，是Feingold-Frenkel代数$\mathfrak{F}$，该方法基于在降维到李代数之前考虑一级状态的张量代数，通过将张量积转换为多重对易子。 这种方法使我们能够利用相互对易的余类Virasoro代数的存在，其数量随着仿射层次的增加而无限增长。 我们给出了所有层次$\ell\leq 5$下张量代数在仿射和余类Virasoro对称性下的完整分解，以及从最大张量基态出发，通过仿射和余类Virasoro生成元的联合作用，以及随后转换为多对易子，从而可以（冗余地）生成$\mathfrak{F}$中直到第五层次的所有元素，这些多对易子则用横向和纵向DDF态表示。 我们概述了未来工作的新方向。 显示英文摘要

We propose a novel approach to study hyperbolic Kac-Moody algebras, and more specifically, the Feingold-Frenkel algebra $\mathfrak{F}$, which is based on considering the tensor algebra of level-one states before descending to the Lie algebra by converting tensor products into multiple commutators. This method enables us to exploit the presence of mutually commuting coset Virasoro algebras, whose number grows without bound with increasing affine level. We present the complete decomposition of the tensor algebra under the affine and coset Virasoro symmetries for all levels $\ell\leq 5$, as well as the maximal tensor ground states from which all elements of $\mathfrak{F}$ up to level five can be (redundantly) generated by the joint action of the affine and coset Virasoro generators, and subsequent conversion to multi-commutators, which are then expressed in terms of transversal and longitudinal DDF states. We outline novel directions for future work.

[7] arXiv:2508.03816 (交叉列表自 math.AG) [中文pdf, pdf, 其他]

标题： 比较辫子簇上的簇代数 显示英文标题

标题： Comparing cluster algebras on braid varieties

Roger Casals, Pavel Galashin, Mikhail Gorsky, Linhui Shen, Melissa Sherman-Bennett, José Simental

评论： 67页，20图

主题： 代数几何 (math.AG) ; 组合数学 (math.CO) ; 表示理论 (math.RT) ; 辛几何 (math.SG)

辫簇参数化由正辫子决定的横截条件的旗的线性配置。 它们包括并推广了约简双布劳特单元、正单纯簇、开放博特-萨姆森簇和理查德森簇等。 最近，在辫簇的坐标环中独立构造了两种簇代数结构：一种使用编织，另一种使用德奥德拉几何。 文章的主要结果是这两种簇代数是一致的。 更一般地，我们的比较研究将每种方法中的不同概念和结果相互对应，涵盖了组合和代数几何方面。 显示英文摘要

Braid varieties parametrize linear configurations of flags with transversality conditions dictated by positive braids. They include and generalize reduced double Bruhat cells, positroid varieties, open Bott-Samelson varieties, and Richardson varieties, among others. Recently, two cluster algebra structures were independently constructed in the coordinate rings of braid varieties: one using weaves and the other using Deodhar geometry. The main result of the article is that these two cluster algebras coincide. More generally, our comparative study matches the different concepts and results from each approach to the other, both on the combinatorial and algebraic geometric aspects.

[8] arXiv:2508.04521 (交叉列表自 math.FA) [中文pdf, pdf, html, 其他]

标题： 二维中的小波协轨道空间分类 显示英文标题

标题： Classifying Wavelet Coorbit Spaces in Dimension 2

Noufal Asharaf, Hartmut Führ, Vaishakh Jayaprakash

主题： 泛函分析 (math.FA) ; 表示理论 (math.RT)

共轨空间为评估广义小波系统的逼近理论性质提供了严格的框架。 因此，了解两种不同的小波系统何时产生相同的共轨空间尺度是有用的。 本文针对二维矩阵群相关的连续小波变换情况，对此问题提供了详尽的答案。 显示英文摘要

Coorbit spaces provide a rigorous framework for the assessment of the approximation theoretic properties of generalized wavelet systems. It is therefore useful to understand when two different wavelet systems give rise to the same scales of coorbit spaces. This paper provides an exhaustive answer to this question for the case of continuous wavelet transforms associated with matrix groups in dimension two.

[9] arXiv:2508.04532 (交叉列表自 math.QA) [中文pdf, pdf, html, 其他]

标题： 伪$q$-迹与(余)端有何关系？ 显示英文标题

标题： How are pseudo-$q$-traces related to (co)ends?

Bin Gui, Hao Zhang

评论： 64页，许多图表

主题： 量子代数 (math.QA) ; 数学物理 (math-ph) ; 表示理论 (math.RT)

设$\mathbb V$是一个$\mathbb N$-分次的$C_2$-有限顶点算子代数（VOA），不一定有理或自对偶。 利用[GZ25a]中缝合-分解定理的一个特例，我们证明了在$\mathrm{Mod}(\mathbb{V}^{\otimes2})$中的端点$\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$（其中$\mathbb{M}'$是$\mathbb{M}$的对偶模）具有与它的$\mathbb{V}^{\otimes2}$-模结构相容的结合$\mathbb C$-代数的自然结构。 Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. 结合这些结果与[GZ25b]中的主定理，我们证明了Gainutdinov-Runkel的一个猜想：对于$\mathbb G$中的任何投射生成器，在$\mathrm{Mod}(\mathbb V)$中，伪$q$-迹构造从$\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$映射到$\mathbb V$的真空环面共形块空间的线性同构。 特别是，如果$A$是一个单位有限维$\mathbb C$-代数，使得有限维左$A$-模的范畴与$\mathrm{Mod}(\mathbb V)$等价，那么$\mathrm{SLF}(A)$与$\mathbb V$的真空环面共形块空间线性同构。这证实了Arike-Nagatomo的一个猜想。 显示英文摘要

Let $\mathbb V$ be an $\mathbb N$-graded $C_2$-cofinite vertex operator algebra (VOA), not necessarily rational or self-dual. Using a special case of the sewing-factorization theorem from [GZ25a], we show that the end $\mathbb E=\int_{\mathbb M\in\mathrm{Mod}(\mathbb V)}\mathbb M\otimes_{\mathbb C}\mathbb M'$ in $\mathrm{Mod}(\mathbb{V}^{\otimes2})$ (where $\mathbb{M}'$ is the contragredient module of $\mathbb{M}$) admits a natural structure of associative $\mathbb C$-algebra compatible with its $\mathbb{V}^{\otimes2}$-module structure. Moreover, we show that a suitable category $\mathrm{Coh}_{\mathrm{L}}(\mathbb E)$ of left $\mathbb E$-modules is isomorphic, as a linear category, to $\mathrm{Mod}(\mathbb V)$, and that the space of vacuum torus conformal blocks is isomorphic to the space $\mathrm{SLF}(\mathbb E)$ of symmetric linear functionals on $\mathbb E$. Combining these results with the main theorem of [GZ25b], we prove a conjecture of Gainutdinov-Runkel: For any projective generator $\mathbb G$ in $\mathrm{Mod}(\mathbb V)$, the pseudo-$q$-trace construction yields a linear isomorphism from $\mathrm{SLF}(\mathrm{End}_{\mathbb V}(\mathbb{G})^{\mathrm{opp}})$ to the space of vacuum torus conformal blocks of $\mathbb V$. In particular, if $A$ is a unital finite-dimensional $\mathbb C$-algebra such that the category of finite-dimensional left $A$-modules is equivalent to $\mathrm{Mod}(\mathbb V)$, then $\mathrm{SLF}(A)$ is linearly isomorphic to the space of vacuum torus conformal blocks of $\mathbb V$. This confirms a conjecture of Arike-Nagatomo.

[10] arXiv:2508.04579 (交叉列表自 math.AG) [中文pdf, pdf, html, 其他]

标题： 类型为$D_4$的简单翻转的倾斜丛导出等价 显示英文标题

标题： Derived equivalence for the simple flop of type $D_4$ via tilting bundles

Wahei Hara

评论： 22页，1张表格，欢迎提出意见

主题： 代数几何 (math.AG) ; 表示理论 (math.RT)

本文的目的是讨论由Kanemitsu发现的类型为$D_4$的简单翻转局部模型的导出等价问题。 首先，在翻转的两侧构造了倾斜丛，然后将这些倾斜丛用于证明导出等价性。 这种翻转的导出等价性推导出一般次数为$12$的K3曲面之间的导出等价性。 对该翻转例子的研究与作者之前对类型为$G_2^{\dagger}$的简单翻转的工作非常相似，但倾斜丛的构造和分析变得更加困难。 显示英文摘要

The aim of this article is to discuss the derived equivalence problem for a local model of the simple flop of type $D_4$, which was found by Kanemitsu. First, tilting bundles on both sides of the flop are constructed, and then those tilting bundles are applied to prove the derived equivalence. This derived equivalence for the flop deduces derived equivalences between general K3 surfaces of degree $12$. The study of this example of a flop is very similar to the author's previous work for the simple of flop of type $G_2^{\dagger}$, but the construction and the analysis of tilting bundles become harder.

[11] arXiv:2508.04624 (交叉列表自 math.AC) [中文pdf, pdf, html, 其他]

标题： 无限多项式环上的对称模 I：幂零商 显示英文标题

标题： Symmetric modules over the infinite polynomial ring I: nilpotent quotients

Rohit Nagpal, Andrew Snowden, Teresa Yu

评论： 40页

主题： 交换代数 (math.AC) ; 表示理论 (math.RT)

科恩证明了无限变量多项式环$R=k[x_1,x_2,\ldots]$在无限对称群$\mathfrak{S}$的作用下是诺特环。 前两位作者开始了一个项目，以详细理解$\mathfrak{S}$-等变的$R$代数。 在之前的工作中，他们分类了$\mathfrak{S}$-素理想 of$R$。 一个重要的$\mathfrak{S}$-素理想的例子是由变量的$(s+1)$次幂生成的理想$\mathfrak{h}_s$。在本文中，我们研究了$R/\mathfrak{h}_s$-模的范畴。我们得到了一些结果，并在这里仅提及三个：(a) 我们确定了该范畴的格罗滕迪克群；(b) 我们证明了克鲁尔-加布里埃尔维数为$s$；以及(c) 我们获得了导出范畴的生成元。本文将在后续工作中起到关键作用，其中我们将研究一般的模。 显示英文摘要

Cohen proved that the infinite variable polynomial ring $R=k[x_1,x_2,\ldots]$ is noetherian with respect to the action of the infinite symmetric group $\mathfrak{S}$. The first two authors began a program to understand the $\mathfrak{S}$-equivariant algebra of $R$ in detail. In previous work, they classified the $\mathfrak{S}$-prime ideals of $R$. An important example of an $\mathfrak{S}$-prime is the ideal $\mathfrak{h}_s$ generated by $(s+1)$st powers of the variables. In this paper, we study the category of $R/\mathfrak{h}_s$-modules. We obtain a number of results, and mention just three here: (a) we determine the Grothendieck group of the category; (b) we show that the Krull--Gabriel dimension is $s$; and (c) we obtain generators for the derived category. This paper will play a key role in subsequent work where we study general modules.

[12] arXiv:2508.04631 (交叉列表自 math.AG) [中文pdf, pdf, html, 其他]

标题： 阿贝尔霍尔范畴 显示英文标题

标题： Abelian Hall categories

Sabin Cautis

主题： 代数几何 (math.AG) ; 数学物理 (math-ph) ; 表示理论 (math.RT)

将一个箭形图与一个有限长度的单子阿贝尔范畴相关联，该范畴对应该 Varagnolo-Vasserot 的相应预投射 K 理论霍尔代数进行分类。 此范畴中的单对象为霍尔代数提供了（对偶）规范基。 特别是，如果箭形图是仿射的，这将为相应的量子环面代数的正部分提供一个基。 我们还证明，这个阿贝尔范畴自然地配备了重归一化的 r 矩阵。 显示英文摘要

To a quiver we associate a finite length monoidal abelian category which categorifies the corresponding preprojective K-theoretic Hall algebra of Varagnolo-Vasserot. The simples in this category provide a (dual) canonical basis of the Hall algebra. In particular, if the quiver is affine, this provides a basis for the positive half of the corresponding quantum toroidal algebra. We also show that this abelian category is naturally endowed with renormalized r-matrices.

替换提交 (展示 8 之 8 条目 )

[13] arXiv:2405.19042 (替换) [中文pdf, pdf, 其他]

标题： $(d+2)$-角范畴上的秩函数 -- 一种函子方法 显示英文标题

标题： Rank functions on $(d+2)$-angulated categories -- a functorial approach

David Nkansah

评论： 31页。通过加强定义2.1（公理RO2）修正了命题2.12。在第2节中添加了例子

主题： 表示理论 (math.RT) ; 范畴论 (math.CT) ; 环与代数 (math.RA)

我们引入了在$(d+2)$-角范畴$\mathcal{C}$上的秩函数的概念，该概念推广了在三角范畴上的秩函数概念。 受Chuang和Lazarev工作的启发，对于$d$为奇正整数的情况，我们证明了在$\mathcal{C}$中对象上定义的秩函数与在$\mathcal{C}$中态射上定义的秩函数之间存在双射对应关系。 受Conde、Gorsky、Marks和Zvonareva工作的启发，对于$d$为一个奇正整数，我们证明了$\operatorname{\mathsf{Proj}}A$上的秩函数与$\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$上的可加函数之间存在双射对应关系，其中$\operatorname{\mathsf{Proj}}A$被赋予了Amiot-Lin$(d+2)$-角范畴结构。这使我们能够证明$\operatorname{\mathsf{Proj}}A$上的每个整数秩函数都可以分解为不可约秩函数。 显示英文摘要

We introduce the notion of a rank function on a $(d+2)$-angulated category $\mathcal{C}$ which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for $d$ an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in $\mathcal{C}$ and rank functions defined on morphisms in $\mathcal{C}$. Inspired by work of Conde, Gorsky, Marks and Zvonareva, for $d$ an odd positive integer, we show there is a bijective correspondence between rank functions on $\operatorname{\mathsf{Proj}}A$ and additive functions on $\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$, where $\operatorname{\mathsf{Proj}}A$ is endowed with the Amiot-Lin $(d+2)$-angulated category structure. This allows us to show that every integral rank function on $\operatorname{\mathsf{Proj}}A$ can be decomposed into irreducible rank functions.

[14] arXiv:2406.10726 (替换) [中文pdf, pdf, html, 其他]

标题： 整数二次型和线性无关根的子集的扩展 显示英文标题

标题： Integer quadratic forms and extensions of subsets of linearly independent roots

Rafael Stekolshchik

评论： 29页，14图，2表，新增第1.1节，第1.11节。 arXiv管理员注释：与arXiv:1406.3049文本重叠

期刊参考： 代数离散数学。39:2 (2025) 243-283

主题： 表示理论 (math.RT)

我们考虑某个根系$\varPhi$中线性无关根的子集。令$S'$为此类子集，令$S'$与任何 Carter 图$\Gamma'$相关。论文的主要问题是：可以向$S'$添加哪个根$\gamma \in \varPhi$，使得$S' \cup \gamma$也是线性无关根的子集？ This extra root $\gamma$ is called the linkage root. The vector $\gamma^{\nabla}$ of inner products $\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}$ is called the linkage label vector. Let $B_{\Gamma'}$ be the Cartan matrix associated with $\Gamma'$. 显示当且仅当 $\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2$时， $\gamma$ 是一个联结根，其中 $\mathscr{B}^{\vee}_{\Gamma'}$ 是一个二次型，其矩阵是 $B_{\Gamma'}$的逆矩阵。 所有 $\Gamma'$ 的联结根的集合称为一个联结系统，并记为 $\mathscr{L}(\Gamma')$。 与任何卡特图$\Gamma'$相关的卡特矩阵与某个迪克森图$\Gamma$相关的卡特矩阵共轭，[St23]。 $\mathscr{L}(\Gamma')$和$\mathscr{L}(\Gamma)$的大小相同。 设$W^{\vee}$为二次型$\mathscr{B}^{\vee}_{\Gamma'}$的威尔群。 该群作用在连接系统上并形成多个轨道。 链接系统$\mathscr{L}(D_l)$和$\mathscr{L}(D_l(a_k))$的轨道大小和结构如上所述。 显示英文摘要

We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $\Gamma'$. The main question of the paper: what root $\gamma \in \varPhi$ can be added to $S'$ so that $S' \cup \gamma$ is also a subset of linearly independent roots? This extra root $\gamma$ is called the linkage root. The vector $\gamma^{\nabla}$ of inner products $\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}$ is called the linkage label vector. Let $B_{\Gamma'}$ be the Cartan matrix associated with $\Gamma'$. It is shown that $\gamma$ is a linkage root if and only if $\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{\Gamma'}$ is a quadratic form with the matrix inverse to $B_{\Gamma'}$. The set of all linkage roots for $\Gamma'$ is called a linkage system and is denoted by $\mathscr{L}(\Gamma')$. The Cartan matrix associated with any Carter diagram $\Gamma'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $\Gamma$, [St23]. The sizes of $\mathscr{L}(\Gamma')$ and $\mathscr{L}(\Gamma)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{\Gamma'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented.

[15] arXiv:2502.15417 (替换) [中文pdf, pdf, 其他]

标题： $τ$-局部代数上箭图表示的例外序列 显示英文标题

标题： $τ$-exceptional sequences for representations of quivers over local algebras

Iacopo Nonis

评论： 39页。v2：删除了命题5.2和推论5.12。一些小的修改

主题： 表示理论 (math.RT)

设$k$为一个代数闭域。 设$R$为一个有限维交换局部$k$-代数，设$Q$为一个无有向圈的图。 在本文中，我们研究代数$\Lambda = R\otimes kQ$上的（带符号）$\tau$-例外序列，该代数与$RQ$同构。 我们证明在$\text{mod }kQ$中的完整(带符号)$\tau$-例外序列集合与$\text{mod }\Lambda$中的完整(带符号)$\tau$-例外序列集合之间存在双射。 此外，我们证明了 $\tau$-正交子范畴的 $\text{mod }\Lambda$与 $R\otimes kQ'$的模范畴等价，其中 $Q'$是某个箭图。 作为推论，我们证明了 $\tau$-簇映射范畴的 $kQ$和 $\Lambda$是等价的。 显示英文摘要

Let $k$ be an algebraically closed field. Let $R$ be a finite dimensional commutative local $k$-algebra and let $Q$ be a quiver with no oriented cycles. In this paper, we study (signed) $\tau$-exceptional sequences over the algebra $\Lambda = R\otimes kQ$, which is isomorphic to $RQ$. We show there is a bijection between the set of complete (signed) $\tau$-exceptional sequences in $\text{mod }kQ$ and the set of complete (signed) $\tau$-exceptional sequences in $\text{mod }\Lambda$. Moreover, we prove that every $\tau$-perpendicular subcategory of $\text{mod }\Lambda$ is equivalent to the module category of $R\otimes kQ'$, for some quiver $Q'$. As a consequence, we prove that the $\tau$-cluster morphism categories of $kQ$ and $\Lambda$ are equivalent.

[16] arXiv:2502.20711 (替换) [中文pdf, pdf, html, 其他]

标题： 有限型Artin-Tits群中的周期元素和稳定性条件 显示英文标题

标题： Periodic elements in finite type Artin-Tits groups and stability conditions

Edmund Heng, Anthony M. Licata, Oded Yacobi

评论： 11页；最终版本将发表在IMRN上

主题： 表示理论 (math.RT) ; 群论 (math.GR) ; 几何拓扑 (math.GT)

有限类型阿廷-蒂茨群中的周期元素是某些正幂次为中心的元素。 我们通过其在相应2-卡比-亚乌范畴及其空间上的（融合等变）布里奇兰德稳定性条件的作用来动态地刻画周期元素。 主要定理是，元素$\beta$是周期的当且仅当$\beta$在稳定性流形上有一个不动点。 显示英文摘要

Periodic elements in finite type Artin--Tits groups are elements some positive power of which is central. We give a dynamical characterisation of periodic elements via their action on the corresponding 2-Calabi--Yau category and on its space of (fusion equivariant) Bridgeland stability conditions. The main theorem is that an element $\beta$ is periodic if and only if $\beta$ has a fixed point in the stability manifold.

[17] arXiv:2507.22836 (替换) [中文pdf, pdf, html, 其他]

标题： 通过奇点理论的简单李代数的几何模型 显示英文标题

标题： Geometric models of simple Lie algebras via singularity theory

Cheol-Hyun Cho, Wonbo Jeong, Beom-Seok Kim

评论： 44页，26图，参考文献已添加

主题： 表示理论 (math.RT) ; 几何拓扑 (math.GT) ; 辛几何 (math.SG)

众所周知，ADE型Dynkin图同时对应该类简单Lie代数和简单奇点。 我们为每个ADE情况引入一个平面多边形轮，称为Coxeter轮。 我们证明Coxeter轮的边和辐条的等价类形成一个与相应类型的经典根系同构的几何根系。 这个轮实际上来源于对应双变量简单奇点的Milnor纤维，几何根系上的双线性形式是其对称化Seifert形式的负数。 此外，我们使用弧、Seifert形式和奇点理论中的变分算子给出了简单Lie代数的一个完全几何定义。 显示英文摘要

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.

[18] arXiv:2303.06704 (替换) [中文pdf, pdf, html, 其他]

标题： 通过丛代数中的突变组合学的有理Weyl群作用和q-Painleve方程 显示英文标题

标题： Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras

Tetsu Masuda, Naoto Okubo, Teruhisa Tsuda

评论： 29页；它于2025年8月6日进行了重大修订，以更新内容

主题： 精确可解与可积系统 (nlin.SI) ; 表示理论 (math.RT)

一个簇代数是一个由称为变异的箭图（有向图）及其相关的简单有理映射的操作生成的代数结构。 通过使用图组合方法，我们提出了一种系统的方法，从簇代数导出Weyl群的热带，即减法自由有理表示。 我们的结果提供了一个广泛的Weyl群作用类，包括具有代数几何背景的先前已知示例，因此与q-Painleve方程及其高阶扩展相关。 论证的关键要素是与箭图中的循环子图相关的反射的组合方面。 我们还研究了由此获得的离散动力系统的辛结构。 斜对称整数矩阵的规范形式允许我们在保持有理性的同时选择Darboux坐标。 显示英文摘要

A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to derive a tropical, i.e. subtraction-free birational, representation of Weyl groups from cluster algebras. Our results provide an extensive class of Weyl group actions, including previously known examples with algebro-geometric background, and hence are relevant to the q-Painleve equations and their higher-order extensions. Key ingredients of the argument are the combinatorial aspects of the reflection associated with a cycle subgraph in the quiver. We also study symplectic structures of the discrete dynamical systems thus obtained. The normal form of a skew-symmetric integer matrix allows us to choose Darboux coordinates while preserving the birationality.

[19] arXiv:2401.00252 (替换) [中文pdf, pdf, html, 其他]

标题： 簇代数与单调拉格朗日环面 显示英文标题

标题： Cluster algebras and monotone Lagrangian tori

Yunhyung Cho, Myungho Kim, Yoosik Kim, Euiyong Park

评论： v2. 45页；包含审稿人的意见和建议

主题： 辛几何 (math.SG) ; 代数几何 (math.AG) ; 表示理论 (math.RT)

受[GHKK18, FO25]中通过簇代数构造牛顿-奥库诺夫体和环面退化的影响，我们考虑了一族由热带化簇变换的复合相关的复光滑法诺流形$X$的牛顿-奥库诺夫多面体。根据[HK15]的工作，该族中每个牛顿-奥库诺夫多面体$\Delta$对应的环面退化在$\Delta$上产生一个完全可积系统$X$。我们研究了每个完全可积系统拥有单调拉格朗日环面纤维的情况。我们提供了一个充分条件，基于热带整数点和交换矩阵的数据，使得构造的单调拉格朗日环面族包含无限多个单调拉格朗日环面，其中任意两个都不被任何辛同构所关联。通过采用这个准则并利用热带整数点与对偶规范基元素之间的对应关系，我们在除少数情况外的任意类型的旗流形上生成了无限多个不同的单调拉格朗日环面。 显示英文摘要

Motivated by the construction of Newton--Okounkov bodies and toric degenerations via cluster algebras in [GHKK18, FO25], we consider a family of Newton--Okounkov polytopes of a complex smooth Fano variety $X$ related by a composition of tropicalized cluster mutations. According to the work of [HK15], the toric degeneration associated with each Newton--Okounkov polytope $\Delta$ in the family produces a completely integrable system of $X$ over $\Delta$. We investigate circumstances in which each completely integrable system possesses a monotone Lagrangian torus fiber. We provide a sufficient condition, based on the data of tropical integer points and exchange matrices, for the family of constructed monotone Lagrangian tori to contain infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphism. By employing this criterion and exploiting the correspondence between the tropical integer points and the dual canonical basis elements, we generate infinitely many distinct monotone Lagrangian tori on flag manifolds of arbitrary type except in a few cases.

[20] arXiv:2412.15169 (替换) [中文pdf, pdf, html, 其他]

标题： 通过范畴化sl(2)作用的窗口等价性 显示英文标题

标题： Window equivalences via categorical sl(2) actions

Wei Tseu

评论： 小的更改

主题： 代数几何 (math.AG) ; 表示理论 (math.RT)

我们通过Knörrer周期性识别了分层Mukai flop的余切丛的导出等价的两种不同方法——一种是由几何范畴sl(2)作用引起的，另一种是通过带电Landau-Ginzburg模型上的分次矩阵因子化的魔法窗口范畴。 显示英文摘要

We identify two distinct approaches to the derived equivalence for the stratified Mukai flop of cotangent bundles of Grassmannians -- one induced by the geometric categorical sl(2) action, and the other through the magic window category of graded matrix factorizations on the gauged Landau-Ginzburg model -- via the Kn\"orrer periodicity.