Advances in Mathematics最新文献

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A simple proof of the Atkin-O'Brien partition Hecke congruence conjecture for powers of 13 13次幂的Atkin-O'Brien分合Hecke同余猜想的一个简单证明

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-30 DOI: 10.1016/j.aim.2026.110840

Frank Garvan, Zhumagali Shomanov

In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. While they proved the conjecture modulo 13 and 13², a proof for all powers of 13 has remained open. In this paper we provide a simple and complete proof of the conjecture.

1967年，Atkin和O'Brien猜想了包含Hecke算子模为13的幂的配分函数的同余。虽然他们证明了模13和模132的猜想，但13的所有幂的证明仍然是开放的。本文给出了这个猜想的一个简单而完整的证明。

引用次数: 0

An explicit derived McKay correspondence for some complex reflection groups of rank two 一类二阶复反射群的显式推导McKay对应

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-30 DOI: 10.1016/j.aim.2026.110794

Anirban Bhaduri , Yael Davidov , Eleonore Faber , Katrina Honigs , Peter McDonald , C. Eric Overton-Walker , Dylan Spence

<div><div>In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups <math><mi>G</mi><mo>=</mo><mi>G</mi><mo>(</mo><mn>2</mn><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math>, <math><msub><mrow><mi>G</mi></mrow><mrow><mn>12</mn></mrow></msub></math>, <math><msub><mrow><mi>G</mi></mrow><mrow><mn>13</mn></mrow></msub></math>, or <math><msub><mrow><mi>G</mi></mrow><mrow><mn>22</mn></mrow></msub></math>, there is a semiorthogonal decomposition of the following form, where <math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math> are the normalizations of the irreducible components of the branch divisor <math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi></math> and <math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math> are exceptional objects:<math><msup><mrow><mi>D</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>≅</mo><mo>〈</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>,</mo><mi>D</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi><mo>)</mo><mo>〉</mo><mo>.</mo></math> We verify that the pieces of this decomposition correspond to the irreducible representations of G, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group <math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math>, this conjecture is now proven for all finite groups <math><mi>G</mi><mo>≤</mo><mi>GL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math> that are generated by order 2 reflections. Each of these groups contains, as a subgroup of index 2, a distinct finite group <math><mi>H</mi><mo>≤</mo><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math>. A key part of our work is an explicit computation of the action of <math><mi>G</mi><mo>/</mo><mi>H</mi></math> on the H-Hilbert scheme <math><mrow><mtext>H</mtext><mtext>-Hilb</mtext></mrow

本文研究了几种反射群的McKay对应，即由二阶反射生成的二阶反射群。我们证明了对于每一个反射群G=G(2m,m,2), G12, G13，或G22，存在如下形式的半正交分解，其中B1，…，Br是分支因子C2→C2/G的不可约分量的归一化，E1，…，En是例外对象：DG(C2) = < E1，…，En，D(B1)，…，D(Br)，D(C2/G) >。我们证明了这个分解的片段对应于G的不可约表示，验证了Polishchuk和Van den Bergh的轨道半正交分解猜想。由于Potter对群G（m,m,2）的研究，这个猜想现在被证明适用于所有由2阶反射产生的有限群G≤GL（2，C）。这些群中的每一个都包含一个不同的有限群H≤SL(2，C)，作为指标2的子群。我们工作的一个关键部分是G/H对H- hilbert方案H- hilb （C2）的显式计算。

引用次数: 0

Non-perturbative localization for quasi-periodic Jacobi block matrices 拟周期Jacobi块矩阵的非微扰局部化

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-30 DOI: 10.1016/j.aim.2026.110832

Rui Han , Wilhelm Schlag

We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori

T^{b}

is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for

b = 1

, and applications to skew shifts, stacked graphene, XY spin chains, and coupled Harper models are presented.

我们证明了假设所有Lyapunov指数不消失的拟周期Jacobi块矩阵算子的非微扰Anderson局域化。假定环面Tb上的基动力学为丢番图旋转。得到了b=1时的算法定位结果，并给出了在歪斜位移、堆叠石墨烯、XY自旋链和耦合Harper模型中的应用。

引用次数: 0

Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities 高斯空间椭圆型偏微分方程解的几何性质及相关的Brunn-Minkowski型不等式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-30 DOI: 10.1016/j.aim.2026.110827

Andrea Colesanti , Lei Qin , Paolo Salani

We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted p-operator

- Δ_{p, γ} u = - div (| \nabla u |^{p - 2} \nabla u) + (x, \nabla u) | \nabla u |^{p - 2},

where

p > 1

, in the class of bounded Lipschitz domains in

R^{n}

. We also prove that the corresponding positive eigenfunctions are log-concave if the domain is convex.

我们证明了Rn中有界Lipschitz区域类中加权p算子- Δp，γu= - div(|∇u|p - 2∇u)+(x,∇u)|∇u|p - 2的第一个（非平凡）Dirichlet特征值的Brunn-Minkowski型不等式，其中p>；1。我们还证明了当定义域为凸时对应的正特征函数是对数凹的。

引用次数: 0

A necessary and sufficient condition for k-transversals k-截线的充分必要条件

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-30 DOI: 10.1016/j.aim.2026.110829

Daniel McGinnis , Nikola Sadovek

We solve a long-standing open problem posed by Goodman & Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in

R^{d}

to admit a k-transversal for any

0 \leq k \leq d - 1

. This result is a common generalization of Helly's theorem (

k = 0

) and the Goodman-Pollack-Wenger theorem (

k = d - 1

). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex k-transversal to a family of convex sets in

C^{d}

, extending the work of McGinnis (

k = d - 1

). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.

我们解决了Goodman &； Pollack（1988）提出的一个长期存在的开放问题，通过建立在Rd上的一组凸集允许任意0≤k≤d−1的k截线的充分必要条件。这个结果是Helly定理（k=0）和Goodman-Pollack-Wenger定理（k=d−1）的一般推广。此外，通过刻画Cd上凸集族的复k-截线的存在性，我们得到了复集合下的一个类似情形，推广了McGinnis （k=d−1）的工作。我们的方法是拓扑的，并在Stiefel流形上使用borsuk - ulam型定理。最后，我们证明了我们的结果如何在真实情况下蕴涵Živaljević-Vrećica和Dol'nikov的中心横截定理，在复杂情况下蕴涵Sadovek-Soberón的中心横截定理。

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引用次数: 0

Integer-valued valuations 整数值的估值

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-29 DOI: 10.1016/j.aim.2026.110823

Andrii Ilienko , Ilya Molchanov , Tommaso Visonà

We obtain a complete characterization of planar monotone σ-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or σ-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.

我们得到了平面单调σ-连续赋值取整数值的完备刻划，且不假设在任何变换群下不变。我们进一步研究了下降单调性或σ-连续性的结果，并给出了线值的完整分类。我们还介绍了这类估值的产品构造。

引用次数: 0

On Galois theory of cluster algebras: general and that from Riemann surfaces 关于簇代数的伽罗瓦理论：一般的和来自黎曼曲面的

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-28 DOI: 10.1016/j.aim.2026.110821

Jinlei Dong, Fang Li

One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois theory, we want to discuss the relations between cluster subalgebras of a cluster algebra and subgroups of its automorphism group and then set up the Galois-like method.

In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra

A

to its cluster automorphism group, and introduce notions of Galois-like extensions and Galois extensions. A necessary condition for Galois extensions of a cluster algebra

A

is given, which is also a sufficient condition if

A

has a

D

-stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups

H_{1}

and

H_{2}

of the automorphism group

Aut A

are conjugate to each other if and only if there exists

f \in Aut A

and two Galois-like extension subalgebras

A (Σ_{1})

A (Σ_{2})

corresponding to

H_{1}

and

H_{2}

such that f is an isomorphism between

A (Σ_{1})

and

A (Σ_{2})

In the second part, as the answers of two conjectures proposed in the first part, for a cluster algebra from a feasible surface, we prove that Galois-like extension subalgebras corresponding to a subgroup of a cluster automorphism group have the same rank. Moreover, it is shown that there are order-preserving reverse Galois maps for these cluster algebras. We also give examples of

D

-stable bases and some discussions on the Galois inverse problem in this part.

伽罗瓦域扩展理论的关键之一是建立域的子域与其自同构群的子群之间的对应关系，从而通过群的方法来研究域。作为伽罗瓦理论的类比，我们讨论了一类聚类代数的聚类子代数与其自同构群的子群之间的关系，并建立了类伽罗瓦方法。在第一部分中，我们建立了从偏对称聚类代数a到它的聚类自同构群的伽罗瓦映射，并引入了类伽罗瓦扩展和伽罗瓦扩展的概念。给出了聚类代数A的伽罗瓦扩展的一个必要条件，同时也是A具有d稳定基或具有唯一表达式的稳定单项式基的充分条件。讨论了类伽罗瓦扩展的一些性质。证明了自同构群AutA的两个子群H1和H2是共轭的当且仅当存在f∈AutA和对应于H1和H2的两个类伽罗司扩展子代数A（Σ1）， A（Σ2），使得f是A（Σ1）和A（Σ2）之间的同构。在第二部分中，作为第一部分中两个猜想的答案，我们从可行曲面上证明了簇自同构群的子群对应的类伽罗瓦扩展子代数具有相同的秩。此外，还证明了这些簇代数存在保序的逆伽罗瓦映射。在这一部分中，我们还给出了d稳定基的例子，并对伽罗瓦逆问题进行了一些讨论。

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引用次数: 0

Quantitative concatenation for polynomial box norms 多项式盒范数的定量拼接

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-28 DOI: 10.1016/j.aim.2026.110820

Noah Kravitz , Borys Kuca , James Leng

Using PET and quantitative concatenation techniques, we establish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper [20], we complete this program for sets in

{[N]}^{2}

lacking nondegenerate progressions of the form

(x, y), (x + P (z), y), (x, y + P (z))

, where

P \in Z [z]

is any fixed polynomial with an integer root of multiplicity 1.

使用PET和定量连接技术，我们建立了具有“期望”方向的盒范数控制，用于一般多维多项式级数的计数算子，参数中最多有多项式损失。这些结果通常是在缺乏给定级数实例的集合上得到显式上界的有用的第一步。在同伴论文[20]中，我们完成了对于[N]2中缺乏形式为（x,y），（x+P(z),y），（x,y+P(z)）的非退化级数的集合的这个规划，其中P∈z [z]是任何复数根为1的固定多项式。

引用次数: 0

Asymptotics for t-core partitions and Stanton's conjecture t核分区的渐近性与Stanton猜想

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-28 DOI: 10.1016/j.aim.2026.110805

Matt Tyler

A partition is a t-core partition if t is not one of its hook lengths. Let

c_{t} (N)

be the number of t-core partitions of N. In 1999, Stanton conjectured

c_{t} (N) \leq c_{t + 1} (N)

4 \leq t \neq N - 1

. This was proved for t fixed and N sufficiently large by Anderson, and for small values of t by Kim and Rouse. In this paper, we prove Stanton's conjecture in general.

Our approach is to find a saddle point asymptotic formula for

c_{t} (N)

, valid in all ranges of t and N. This includes the known asymptotic formulas for

c_{t} (N)

as special cases, and shows that the behavior of

c_{t} (N)

depends on how

t^{2}

compares in size to N. For example, our formula implies that if

t^{2} = κ N + o (t)

, then

c_{t} (N) = \frac{\exp (2 π \sqrt{A N})}{B N} (1 + o (1))

for suitable constants A and B defined in terms of κ.

如果t不是其钩子长度之一，则分区为t核分区。设ct(N)为N的t核分区数。1999年，Stanton推测如果4≤t≠N−1，则ct(N)≤ct+1(N)。对于t固定且N足够大的情况，安德森证明了这一点，对于t很小的情况，金和劳斯也证明了这一点。本文一般地证明了斯坦顿猜想。我们的方法是找到ct(N)的鞍点渐近公式，在t和N的所有范围内都有效，这包括已知的ct(N)的渐近公式作为特殊情况，并表明ct(N)的行为取决于t2的大小与N的比较。例如，我们的公式表明，如果t2=κN+o(t)，那么对于κ定义的合适常数a和B， ct(N)=exp (2πAN)BN（1+o(1)）。

引用次数: 0

Exact dg categories I: Foundations 确切的dg类别1：基金会

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-01-28 DOI: 10.1016/j.aim.2026.110809

Xiaofa Chen

We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka–Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the

H^{0}

-category of an exact dg category

A

is triangulated if and only if

A

is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.

我们引入了精确dg范畴的概念，它是对Nakaoka-Palu的外三角化范畴概念的微分分级强化。我们给出了一个与Quillen的定义完全相似的定义，但是用一个更复杂的同伦范畴代替了核-核对的范畴。引入稳定dg范畴的概念，证明了精确dg范畴A的h -范畴是三角化的当且仅当A是稳定的。我们用几个例子来说明我们的理论，包括两项复合体的同伦范畴和广义簇范畴的Amiot基本定义域。

引用次数: 0

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