Advances in Mathematics最新文献

{"title":"Rigidity of the Delaunay triangulations of the plane","authors":"Song Dai ,&nbsp;Tianqi Wu","doi":"10.1016/j.aim.2024.109910","DOIUrl":"10.1016/j.aim.2024.109910","url":null,"abstract":"<div><p>We prove a rigidity result for Delaunay triangulations of the plane under Luo's notion of discrete conformality, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We follow Zhengxu He's analytical approach in his work on the rigidity of disk patterns, and develop a discrete Schwarz lemma and a discrete Liouville theorem. As a key ingredient to prove the discrete Schwarz lemma, we establish a correspondence between the Euclidean discrete conformality and the hyperbolic discrete conformality, for geodesic embeddings of triangulations. Other major tools include conformal modulus, discrete extremal length, and maximum principles in discrete conformal geometry.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087957","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":"Quantum Sugawara operators in type A","authors":"Naihuan Jing ,&nbsp;Ming Liu ,&nbsp;Alexander Molev","doi":"10.1016/j.aim.2024.109907","DOIUrl":"10.1016/j.aim.2024.109907","url":null,"abstract":"<div><p>The quantum Sugawara operators associated with a simple Lie algebra <span><math><mi>g</mi></math></span> are elements of the center of a completion of the quantum affine algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mover><mrow><mi>g</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> at the critical level. By the foundational work of Reshetikhin and Semenov-Tian-Shansky (1990), such operators occur as coefficients of a formal Laurent series <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>V</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span> associated with every finite-dimensional representation <em>V</em> of the quantum affine algebra. As demonstrated by Ding and Etingof (1994), the quantum Sugawara operators generate all singular vectors in generic Verma modules over <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mover><mrow><mi>g</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> at the critical level and give rise to a commuting family of transfer matrices. Furthermore, as observed by E. Frenkel and Reshetikhin (1999), the operators are closely related with the <em>q</em>-characters and <em>q</em>-deformed <span><math><mi>W</mi></math></span>-algebras via the Harish-Chandra homomorphism.</p><p>We produce explicit quantum Sugawara operators for the quantum affine algebra of type <em>A</em> which are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This opens a way to understand all the related objects via their explicit constructions. We consider one application by calculating the Harish-Chandra images of the quantum Sugawara operators. The operators act by scalar multiplication in the <em>q</em>-deformed Wakimoto modules and we calculate the eigenvalues by identifying them with the Harish-Chandra images.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142089295","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":"Growing integer partitions with uniform marginals and the equivalence of partition ensembles","authors":"Yuri Yakubovich","doi":"10.1016/j.aim.2024.109908","DOIUrl":"10.1016/j.aim.2024.109908","url":null,"abstract":"<div><p>We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level <em>n</em>, it passes through any partition <em>λ</em> of <em>n</em> with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.</p><p>Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of <em>n</em> are asymptotically independent as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span> and to find their limiting distributions, which are, somewhat surprisingly, different.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142089654","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":"The extremals of the Kahn-Saks inequality","authors":"Ramon van Handel ,&nbsp;Alan Yan ,&nbsp;Xinmeng Zeng","doi":"10.1016/j.aim.2024.109892","DOIUrl":"10.1016/j.aim.2024.109892","url":null,"abstract":"<div><p>A classical result of Kahn and Saks states that given any partially ordered set with two distinguished elements, the number of linear extensions in which the ranks of the distinguished elements differ by <em>k</em> is log-concave as a function of <em>k</em>. The log-concave sequences that can arise in this manner prove to exhibit a much richer structure, however, than is evident from log-concavity alone. The main result of this paper is a complete characterization of the extremals of the Kahn-Saks inequality: we obtain a detailed combinatorial understanding of where and what kind of geometric progressions can appear in these log-concave sequences. This settles a partial conjecture of Chan-Pak-Panova, while the analysis uncovers new extremals that were not previously conjectured. The proof relies on a much more general geometric mechanism—a hard Lefschetz theorem for nef classes that was obtained in the setting of convex polytopes by Shenfeld and Van Handel—which forms a model for the investigation of such structures in other combinatorial problems.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083193","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":"A positive/tropical critical point theorem and mirror symmetry","authors":"Jamie Judd,&nbsp;Konstanze Rietsch","doi":"10.1016/j.aim.2024.109911","DOIUrl":"10.1016/j.aim.2024.109911","url":null,"abstract":"<div><p>Call a Laurent polynomial <em>W</em> ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose <em>W</em> is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in <span><math><mi>R</mi></math></span>) is called ‘positive’ if the coefficient of its leading term is in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>&gt;</mo><mn>0</mn></mrow></msub></math></span>. We show that <em>W</em> has a unique <em>positive</em> critical point <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>crit</mi></mrow></msub></math></span>, i.e. all of whose coordinates are positive, if and only if <em>W</em> is complete. For any complete, positive Laurent polynomial <em>W</em> in <em>r</em> variables we also obtain from its positive critical point <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>crit</mi></mrow></msub></math></span> a canonically associated ‘tropical critical point’ <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>crit</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> by considering the valuations of the coordinates of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>crit</mi></mrow></msub></math></span>. Moreover we give a finite recursive construction of <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>crit</mi></mrow></msub></math></span> in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of <em>W</em>.</p><p>We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in <em>W</em> are not integral but in <span><math><mi>R</mi></math></span>, even though <em>W</em> is then no longer Laurent.</p><p>Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>Σ</mi></mrow></msub></math></span> we apply our results to obtain for any divisor class <span><math><mo>[</mo><mi>D</mi><mo>]</mo></math></span> satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>Σ</mi></mrow></msub></math></span> to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of <span><span>[11]</span></span> and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using <span><span>[13]</span></span>, <span><span>[37]</span></span>.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004262/pdfft?md5=00f569597dee36082263eecfeccf51e9&pid=1-s2.0-S0001870824004262-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Renormalization in Lorenz maps - completely invariant sets and periodic orbits","authors":"Łukasz Cholewa ,&nbsp;Piotr Oprocha","doi":"10.1016/j.aim.2024.109890","DOIUrl":"10.1016/j.aim.2024.109890","url":null,"abstract":"<div><p>The paper deals with dynamics of expanding Lorenz maps, which appear in a natural way as Poincarè maps in geometric models of well-known Lorenz attractor. Using both analytical and symbolic approaches, we study connections between periodic points, completely invariant sets and renormalizations. We show that some renormalizations may be connected with completely invariant sets while some others don't. We provide an algorithm to detect the renormalizations that can be recovered from completely invariant sets. Furthermore, we discuss the importance of distinguish one-side and double-side preimage. This way we provide a better insight into the structure of renormalizations in Lorenz maps. These relations remained unnoticed in the literature, therefore we are correcting some gaps existing in the literature, improving and completing to some extent the description of possible dynamics in this important field of study.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142058307","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":"Simple vertex algebras arising from congruence subgroups","authors":"Xuanzhong Dai ,&nbsp;Bailin Song","doi":"10.1016/j.aim.2024.109900","DOIUrl":"10.1016/j.aim.2024.109900","url":null,"abstract":"<div><p>For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142058308","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":"Positivity of Peterson Schubert calculus","authors":"Rebecca Goldin ,&nbsp;Leonardo Mihalcea ,&nbsp;Rahul Singh","doi":"10.1016/j.aim.2024.109879","DOIUrl":"10.1016/j.aim.2024.109879","url":null,"abstract":"<div><p>The Peterson variety is a subvariety of the flag manifold <span><math><mi>G</mi><mo>/</mo><mi>B</mi></math></span> equipped with an action of a one-dimensional torus, and a torus invariant paving by affine cells, called Peterson cells. We prove that the equivariant pull-backs of Schubert classes indexed by arbitrary Coxeter elements are dual (up to an intersection multiplicity) to the fundamental classes of Peterson cell closures. Dividing these classes by the intersection multiplicities yields a <span><math><mi>Z</mi></math></span>-basis for the equivariant cohomology of the Peterson variety. We prove several properties of this basis, including a Graham positivity property for its structure constants, and stability with respect to inclusion in a larger Peterson variety. We also find formulae for intersection multiplicities with Peterson classes. This explains geometrically, in arbitrary Lie type, recent positivity statements proved in type A by Goldin and Gorbutt.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020714","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":"Density of the boundary regular set of 2d area minimizing currents with arbitrary codimension and multiplicity","authors":"Stefano Nardulli ,&nbsp;Reinaldo Resende","doi":"10.1016/j.aim.2024.109889","DOIUrl":"10.1016/j.aim.2024.109889","url":null,"abstract":"<div><p>In the present work, we consider area minimizing currents in the general setting of arbitrary codimension and arbitrary boundary multiplicity. We study the boundary regularity of 2<em>d</em> area minimizing currents, beyond that, several results are stated in the more general context of <span><math><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>-almost area minimizing currents of arbitrary dimension <em>m</em> and arbitrary codimension taking the boundary with arbitrary multiplicity. Furthermore, we do not consider any type of convex barrier assumption on the boundary, in our main regularity result which states that the regular set, which includes one-sided and two-sided points, of any 2<em>d</em> area minimizing current <em>T</em> is an open dense set in the boundary.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004043/pdfft?md5=08b837fb2406fc5bb12de7e3d71a9b9d&pid=1-s2.0-S0001870824004043-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142020554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weighted hyperbolic composition groups on the disc and subordinated integral operators","authors":"Luciano Abadias ,&nbsp;José E. Galé ,&nbsp;Pedro J. Miana ,&nbsp;Jesús Oliva-Maza","doi":"10.1016/j.aim.2024.109877","DOIUrl":"10.1016/j.aim.2024.109877","url":null,"abstract":"<div><p>We provide the spectral picture of groups of weighted composition operators, induced by the hyperbolic group of automorphisms of the unit disc, acting on holomorphic functions. Some questions about the spectrum of single weighted hyperbolic composition operators are discussed, and results related with them in the literature are completed or partly extended. Also, our results on the weighted hyperbolic group are applied to the spectral study of two families of multiparameter weighted averaging operators, which generalize both Siskakis' operator and the reduced Hilbert matrix operator.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S000187082400392X/pdfft?md5=271b58f2616162e837efcf782bdabd66&pid=1-s2.0-S000187082400392X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142012929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}