Advances in Mathematics最新文献

{"title":"Diffusion limit with optimal convergence rate of classical solutions to the Vlasov-Maxwell-Boltzmann system","authors":"Tong Yang ,&nbsp;Mingying Zhong","doi":"10.1016/j.aim.2026.110800","DOIUrl":"10.1016/j.aim.2026.110800","url":null,"abstract":"<div><div>We study the diffusion limit of the classical solution to the Vlasov-Maxwell-Boltzmann (VMB) system with initial data near a global Maxwellian. By introducing a new decomposition of the solution to identify the essential components for generating the initial layer, we prove the convergence and establish the optimal convergence rate of the classical solution to the VMB system to the solution of the Navier-Stokes-Maxwell system based on the spectral analysis.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110800"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New results in analysis of Orlicz-Lorentz spaces","authors":"Luis Bernal-González ,&nbsp;Daniel L. Rodríguez-Vidanes ,&nbsp;Juan B. Seoane-Sepúlveda ,&nbsp;Hyung-Joon Tag","doi":"10.1016/j.aim.2026.110808","DOIUrl":"10.1016/j.aim.2026.110808","url":null,"abstract":"<div><div>In this article, we investigate the existence of closed vector subspaces (i.e. spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub></math></span> equipped with the Luxemburg norm. If a family of Orlicz functions <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> satisfies certain order relations with respect to a given Orlicz function <em>φ</em>, the subset of the order-continuous subspace <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msub></math></span> whose elements do not belong to <span><math><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>Λ</mi></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>w</mi></mrow></msub></math></span> is spaceable, and even maximal-spaceable when <em>φ</em> satisfies the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-spaceable are provided. A similar analysis is also performed on the subset <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>∖</mo><msub><mrow><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msub></math></span> when <em>φ</em> does not satisfy the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition.</div><div>The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span>. For a fixed Orlicz function that satisfies the <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> for which the lineability problem is not valid. Moreover, various types of <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110808"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Upper bound for the free energy of dilute Bose gases at low temperature","authors":"Florian Haberberger ,&nbsp;Christian Hainzl ,&nbsp;Benjamin Schlein ,&nbsp;Arnaud Triay","doi":"10.1016/j.aim.2026.110825","DOIUrl":"10.1016/j.aim.2026.110825","url":null,"abstract":"<div><div>We consider a Bose gas at density <span><math><mi>ρ</mi><mo>&gt;</mo><mn>0</mn></math></span>, interacting through a repulsive potential <span><math><mi>V</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> with scattering length <span><math><mi>a</mi><mo>&gt;</mo><mn>0</mn></math></span>. We prove an upper bound for the free energy of the system, valid at low temperature <span><math><mi>T</mi><mo>≲</mo><mi>ρ</mi><mi>a</mi></math></span>. Combined with the recent lower bound obtained in <span><span>[18]</span></span>, our estimate resolves the free energy per unit volume up to and including the Lee–Huang–Yang order <span><math><mi>a</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>ρ</mi><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110825"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Joint distribution of primes in multiple short intervals","authors":"Sun-Kai Leung","doi":"10.1016/j.aim.2026.110847","DOIUrl":"10.1016/j.aim.2026.110847","url":null,"abstract":"<div><div>Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple short intervals follows a multivariate Gaussian distribution with weak negative correlations. As an application, we obtain short-interval analogues of many results in the literature on the Shanks–Rényi prime number race, including a sharp phase transition: biased races between primes in short intervals emerge once the number of intervals exceeds an explicit critical threshold. Our result is new even for a single moving interval, particularly under a quantitative formulation of the linear independence conjecture (QLI).</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110847"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial ergodic theorems in the spirit of Dunford and Zygmund","authors":"Dariusz Kosz ,&nbsp;Bartosz Langowski ,&nbsp;Mariusz Mirek ,&nbsp;Paweł Plewa","doi":"10.1016/j.aim.2026.110859","DOIUrl":"10.1016/j.aim.2026.110859","url":null,"abstract":"<div><div>The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>, <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of uniform oscillation estimates for multiparameter polynomial Radon averaging operators. In the proof of our main result we develop flexible Fourier methods that exhibit and handle the so-called <em>“parameters-gluing”</em> phenomenon, an obstruction that arises in studying oscillation and variation inequalities for multiparameter polynomial Radon operators. We will also discuss connections of our main result with a multiparameter variant of the Bellow–Furstenberg problem.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110859"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Systolic lattice extensions of classical Schottky groups","authors":"Junzhi Huang ,&nbsp;Matthew Zevenbergen","doi":"10.1016/j.aim.2026.110848","DOIUrl":"10.1016/j.aim.2026.110848","url":null,"abstract":"<div><div>We produce lattice extensions of a dense family of classical Schottky subgroups of the isometry group of <em>d</em>-dimensional hyperbolic space. The extensions produced are said to be systolic, since all loxodromic elements with short translation length are conjugate into the Schottky groups. Various corollaries are obtained, in particular showing that for all <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, the set of complex translation lengths realized by systoles of closed hyperbolic <em>d</em>-manifolds is dense inside the set of all possible complex translation lengths. We also consider complex translation lengths in arithmetic hyperbolic <em>d</em>-manifolds, and provide a new way to construct non-arithmetic lattices.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110848"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Ulam widths of finitely presented infinite simple groups","authors":"James Hyde ,&nbsp;Yash Lodha","doi":"10.1016/j.aim.2026.110824","DOIUrl":"10.1016/j.aim.2026.110824","url":null,"abstract":"<div><div>A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from 1947 is <em>uniform simplicity</em>. A group <em>G</em> is said to be <em>n-uniformly simple</em> for <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> if for every <span><math><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo>∖</mo><mo>{</mo><mi>i</mi><mi>d</mi><mo>}</mo></math></span>, there is a product of no more than <em>n</em> conjugates of <em>g</em> and <span><math><msup><mrow><mi>g</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> that equals <em>f</em>. Then <em>G</em> is <em>uniformly simple</em> if it is <em>n-uniformly simple</em> for some <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>, and we refer to the smallest such <em>n</em> as the <em>Ulam width</em>, denoted as <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. If <em>G</em> is simple but not uniformly simple, one declares <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mo>∞</mo></math></span>. In this article, we construct for each <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>, a finitely presented infinite simple group <em>G</em> such that <span><math><mi>n</mi><mo>&lt;</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>&lt;</mo><mo>∞</mo></math></span>. These are the first such examples among the class of finitely presented infinite simple groups. For the class of finitely generated (but not finitely presentable) infinite simple groups, the existence of such examples was settled in the work of Muranov <span><span>[21]</span></span>. However, this had remained open for the class of finitely presented infinite simple groups. Our examples are also of type <span><math><msub><mrow><mi>F</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, which means that they are fundamental groups of aspherical CW complexes with finitely many cells in each dimension. Uniformly simple groups are in particular <em>uniformly perfect</em>: there is an <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> such that every element of the group can be expressed as a product of at most <em>n</em> commutators of elements in the group. We also show that the analogous notion of width for uniform perfection is unbounded for our family of finitely presented infinite simple groups. To our knowledge, this is also the first such family.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110824"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New Brunn–Minkowski and functional inequalities via convexity of entropy","authors":"Gautam Aishwarya ,&nbsp;Liran Rotem","doi":"10.1016/j.aim.2026.110841","DOIUrl":"10.1016/j.aim.2026.110841","url":null,"abstract":"<div><div>We study the connection between the concavity properties of a measure <em>ν</em> and the convexity properties of the associated relative entropy <span><math><mi>D</mi><mo>(</mo><mo>⋅</mo><mo>‖</mo><mi>ν</mi><mo>)</mo></math></span> along optimal transport. As a corollary we prove a new dimensional Brunn–Minkowski inequality for centered star-shaped bodies, when the measure <em>ν</em> is log-concave with a p-homogeneous potential (such as the Gaussian measure). Our method allows us to go beyond the usual convexity assumption on the sets that is fundamentally essential for the standard differential-geometric technique in this area.</div><div>We then take a finer look at the convexity properties of the Gaussian relative entropy, which yields new functional inequalities. First we obtain curvature and dimensional reinforcements to Otto–Villani's HWI inequality in Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' Logarithmic Sobolev inequality and Talagrand's transportation cost inequality in this setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110841"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An explicit derived McKay correspondence for some complex reflection groups of rank two","authors":"Anirban Bhaduri ,&nbsp;Yael Davidov ,&nbsp;Eleonore Faber ,&nbsp;Katrina Honigs ,&nbsp;Peter McDonald ,&nbsp;C. Eric Overton-Walker ,&nbsp;Dylan Spence","doi":"10.1016/j.aim.2026.110794","DOIUrl":"10.1016/j.aim.2026.110794","url":null,"abstract":"&lt;div&gt;&lt;div&gt;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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;12&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;13&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, or &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;22&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, there is a semiorthogonal decomposition of the following form, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are the normalizations of the irreducible components of the branch divisor &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are exceptional objects:&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; We verify that the pieces of this decomposition correspond to the irreducible representations of &lt;em&gt;G&lt;/em&gt;, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, this conjecture is now proven for all finite groups &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;GL&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; that are generated by order 2 reflections. Each of these groups contains, as a subgroup of index 2, a distinct finite group &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;SL&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. A key part of our work is an explicit computation of the action of &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; on the &lt;em&gt;H&lt;/em&gt;-Hilbert scheme &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtext&gt;H&lt;/mtext&gt;&lt;mtext&gt;-Hilb&lt;/mtext&gt;&lt;/mrow","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110794"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Galois theory of cluster algebras: general and that from Riemann surfaces","authors":"Jinlei Dong,&nbsp;Fang Li","doi":"10.1016/j.aim.2026.110821","DOIUrl":"10.1016/j.aim.2026.110821","url":null,"abstract":"<div><div>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.</div><div>In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra <span><math><mi>A</mi></math></span> 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 <span><math><mi>A</mi></math></span> is given, which is also a sufficient condition if <span><math><mi>A</mi></math></span> has a <span><math><mi>D</mi></math></span>-stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of the automorphism group <span><math><mtext>Aut</mtext><mi>A</mi></math></span> are conjugate to each other if and only if there exists <span><math><mi>f</mi><mo>∈</mo><mtext>Aut</mtext><mi>A</mi></math></span> and two Galois-like extension subalgebras <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>, <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> corresponding to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> such that <em>f</em> is an isomorphism between <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>.</div><div>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 <span><math><mi>D</mi></math></span>-stable bases and some discussions on the Galois inverse problem in this part.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110821"},"PeriodicalIF":1.5,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}