Pub Date : 2022-07-03DOI: 10.2140/pjm.2023.322.171
Shun Sawabe
We consider the potential function of the colored Jones polynomial for a link with arbitrary colors and obtain the cone-manifold structure for the link complement. In addition, we establish a relationship between a saddle point equation and hyperbolicity of the link complement. This provides evidence for the Chen-Yang conjecture on the link complement.
{"title":"On the potential function of the colored Jones polynomial with arbitrary colors","authors":"Shun Sawabe","doi":"10.2140/pjm.2023.322.171","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.171","url":null,"abstract":"We consider the potential function of the colored Jones polynomial for a link with arbitrary colors and obtain the cone-manifold structure for the link complement. In addition, we establish a relationship between a saddle point equation and hyperbolicity of the link complement. This provides evidence for the Chen-Yang conjecture on the link complement.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45239224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-01DOI: 10.2140/pjm.2023.323.253
William Chang, A. Manion
We equip the basic local crossing bimodules in Ozsv'ath-Szab'o's theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the $U_q(mathfrak{gl}(1|1)^+)$-intertwining property of the corresponding maps between ordinary representations. Besides yielding a new connection between bordered knot Floer homology and higher representation theory in line with work of Rouquier and the second author, this structure gives an algebraic reformulation of a ``compatibility between summands'' property for Ozsv'ath-Szab'o's bimodules that is important when building their theory up from local crossings to more global tangles and knots.
{"title":"Compatibility in Ozsváth–Szabó’s bordered\u0000HFK via higher representations","authors":"William Chang, A. Manion","doi":"10.2140/pjm.2023.323.253","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.253","url":null,"abstract":"We equip the basic local crossing bimodules in Ozsv'ath-Szab'o's theory of bordered knot Floer homology with the structure of 1-morphisms of 2-representations, categorifying the $U_q(mathfrak{gl}(1|1)^+)$-intertwining property of the corresponding maps between ordinary representations. Besides yielding a new connection between bordered knot Floer homology and higher representation theory in line with work of Rouquier and the second author, this structure gives an algebraic reformulation of a ``compatibility between summands'' property for Ozsv'ath-Szab'o's bimodules that is important when building their theory up from local crossings to more global tangles and knots.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41545178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-22DOI: 10.2140/pjm.2023.322.251
Sarah Brauner, Patrick Commins, V. Reiner
We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by D'esarm'enien and Wachs.
{"title":"Invariant theory for the free left-regular band and a q-analogue","authors":"Sarah Brauner, Patrick Commins, V. Reiner","doi":"10.2140/pjm.2023.322.251","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.251","url":null,"abstract":"We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by D'esarm'enien and Wachs.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46396174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"C1-continuation of periodic orbits from\u0000homoclinics","authors":"Chong-qing Cheng, Min Zhou","doi":"10.2140/pjm.2022.317.67","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.67","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45729983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-19DOI: 10.2140/pjm.2022.317.153
M. Melgaard, Minbo Yang, Xianmei Zhou
We consider the positive solutions of some integral systems related to the static Hartree-type equations:
考虑一类与静态hartree型方程相关的积分系统的正解:
{"title":"Regularity, symmetry and asymptotic behaviour\u0000of solutions for some Stein–Weiss-type integral systems","authors":"M. Melgaard, Minbo Yang, Xianmei Zhou","doi":"10.2140/pjm.2022.317.153","DOIUrl":"https://doi.org/10.2140/pjm.2022.317.153","url":null,"abstract":"We consider the positive solutions of some integral systems related to the static Hartree-type equations:","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43001533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct families of embedded, singly periodic Karcher--Scherk saddle towers of any genus $g$ in the quotient with any even number $2n>2$ of almost parallel Scherk ends. A surface in such a family looks like $n$ parallel planes connected by $n-1+g$ small catenoid necks. In the limit, the family converges to an $n$-sheeted vertical plane with $n-1+g$ singular points termed nodes in the quotient. For the nodes to open up into catenoid necks, their locations must satisfy a set of balance equations whose solutions are given by the roots of Stieltjes polynomials. In a subsequent paper, we will construct minimal surfaces by gluing saddle towers with catenoid limits of saddle towers along their wings.
{"title":"Catenoid limits of singly periodic minimal surfaces with Scherk-type ends","authors":"Hao Chen, Peter Connor, Kevin Li","doi":"10.2140/pjm.2023.325.11","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.11","url":null,"abstract":"We construct families of embedded, singly periodic Karcher--Scherk saddle towers of any genus $g$ in the quotient with any even number $2n>2$ of almost parallel Scherk ends. A surface in such a family looks like $n$ parallel planes connected by $n-1+g$ small catenoid necks. In the limit, the family converges to an $n$-sheeted vertical plane with $n-1+g$ singular points termed nodes in the quotient. For the nodes to open up into catenoid necks, their locations must satisfy a set of balance equations whose solutions are given by the roots of Stieltjes polynomials. In a subsequent paper, we will construct minimal surfaces by gluing saddle towers with catenoid limits of saddle towers along their wings.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44793997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-26DOI: 10.2140/pjm.2023.323.337
Alexander Moret'o, Benjamin Sambale
Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.
设G是一个有Sylow P -子群P的有限群，我们证明了G的特征表决定了P是否有极大幂零类和P是否为极小非阿贝尔群。后一种结果是根据其组成因子对相应组G进行精确分类得到的。对于P约束群G，我们进一步证明了字符表决定了P是否可以由两个元素生成。
{"title":"Groups with 2-generated Sylow subgroups and their character tables","authors":"Alexander Moret'o, Benjamin Sambale","doi":"10.2140/pjm.2023.323.337","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.337","url":null,"abstract":"Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44791614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-14DOI: 10.2140/pjm.2023.322.243
S. Belinschi, H. Bercovici, Ching-Wei Ho
It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.
{"title":"Regularity for free multiplicative convolution on the unit circle","authors":"S. Belinschi, H. Bercovici, Ching-Wei Ho","doi":"10.2140/pjm.2023.322.243","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.243","url":null,"abstract":"It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43000342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-12DOI: 10.2140/pjm.2023.324.371
T. Uzuki, M. Angé, Matthias Aschenbrenner, Robert Lipshitz, Paul Balmer, Kefeng Liu, Paul Yang, Vyjayanthi Chari, S. Popa
Iwase and Matsumoto defined `pochette surgery' as a cut-and-paste on 4-manifolds along a 4-manifold homotopy equivalent to $S^2vee S^1$. The first author in [10] studied infinitely many homotopy 4-spheres obtained by pochette surgery. In this paper we compute the homology of pochette surgery of any homology 4-sphere by using `linking number' of a pochette embedding. We prove that pochette surgery with the trivial cord does not change the diffeomorphism type or gives a Gluck surgery. We also show that there exist pochette surgeries on the 4-sphere with a non-trivial core sphere and a non-trivial cord such that the surgeries give the 4-sphere.
Iwase和Matsumoto将“pochette外科”定义为沿着等价于$S^2 vee S^1$的4-流形上的剪切和粘贴。第一作者在[10]中研究了通过波切特运算得到的无穷多个同伦论4-球。本文利用pochette嵌入的“连接数”计算了任意同调4球的pochette运算的同调性。我们证明了用琐碎的脊髓进行波切手术不会改变异型性类型或进行Gluck手术。我们还证明了在4球体上存在波切特手术，该手术具有非平凡的核心球体和非平凡的绳索，使得手术产生4球体。
{"title":"Pochette surgery of 4-sphere","authors":"T. Uzuki, M. Angé, Matthias Aschenbrenner, Robert Lipshitz, Paul Balmer, Kefeng Liu, Paul Yang, Vyjayanthi Chari, S. Popa","doi":"10.2140/pjm.2023.324.371","DOIUrl":"https://doi.org/10.2140/pjm.2023.324.371","url":null,"abstract":"Iwase and Matsumoto defined `pochette surgery' as a cut-and-paste on 4-manifolds along a 4-manifold homotopy equivalent to $S^2vee S^1$. The first author in [10] studied infinitely many homotopy 4-spheres obtained by pochette surgery. In this paper we compute the homology of pochette surgery of any homology 4-sphere by using `linking number' of a pochette embedding. We prove that pochette surgery with the trivial cord does not change the diffeomorphism type or gives a Gluck surgery. We also show that there exist pochette surgeries on the 4-sphere with a non-trivial core sphere and a non-trivial cord such that the surgeries give the 4-sphere.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45726596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-28DOI: 10.2140/pjm.2023.322.341
A. Khartov
We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the L'evy--Khinchine type representations. We are interested in criteria of weak convergence within this class. Under rather natural assumptions, we state assertions, which connect a weak convergence of quasi-infinitely divisible distribution functions with one special type of convergence of their L'evy--Khinchine spectral functions. The latter convergence is not equivalent to the weak convergence. So we complement known results by Lindner, Pan, and Sato (2018) in this field.
{"title":"On weak convergence of quasi-infinitely divisible laws","authors":"A. Khartov","doi":"10.2140/pjm.2023.322.341","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.341","url":null,"abstract":"We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the L'evy--Khinchine type representations. We are interested in criteria of weak convergence within this class. Under rather natural assumptions, we state assertions, which connect a weak convergence of quasi-infinitely divisible distribution functions with one special type of convergence of their L'evy--Khinchine spectral functions. The latter convergence is not equivalent to the weak convergence. So we complement known results by Lindner, Pan, and Sato (2018) in this field.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44873139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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