Pub Date : 2022-01-31DOI: 10.1017/S0017089522000118
D. Kishimoto, Masahiro Takeda, Yichen Tong
Abstract Ganea proved that the loop space of $mathbb{C} P^n$ is homotopy commutative if and only if $n=3$ . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.
{"title":"Homotopy commutativity in Hermitian symmetric spaces","authors":"D. Kishimoto, Masahiro Takeda, Yichen Tong","doi":"10.1017/S0017089522000118","DOIUrl":"https://doi.org/10.1017/S0017089522000118","url":null,"abstract":"Abstract Ganea proved that the loop space of \u0000$mathbb{C} P^n$\u0000 is homotopy commutative if and only if \u0000$n=3$\u0000 . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but \u0000$mathbb{C} P^3$\u0000 are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds \u0000$G/T$\u0000 for a maximal torus T of a compact, connected Lie group G.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"746 - 752"},"PeriodicalIF":0.5,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44703460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-30DOI: 10.1017/S0017089523000071
Anbu Arjunan, Sruthymurali, S. Sundar
Abstract Let $C_c^{*}(mathbb{N}^{2})$ be the universal $C^{*}$ -algebra generated by a semigroup of isometries ${v_{(m,n)},:, m,n in mathbb{N}}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(mathbb{N}^2)$ for the time evolution determined by a homomorphism $c,:,mathbb{Z}^{2} to mathbb{R}$ . In contrast to the reduced version $C_{red}^{*}(mathbb{N}^{2})$ , we show that the set of KMS states on $C_{c}^{*}(mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.
摘要设$C_C^｛*｝（mathbb｛N｝^｛2｝）$是区间投影可交换的等距$｛v_｛（m，N）｝，：，m，Ninmathbb{N｝｝$的半群生成的泛$C^｛*}$代数。我们分析了由同态$C，：，mathbb｛Z｝^｛2｝ to mathbb{R｝$确定的时间演化的$C_｛C｝^｝*｝（mathbb｝N｝^2）$上的KMS态的结构。与简化版本$C_｛red｝^｛*｝（mathbb｛N｝^{2｝）$相反，我们证明了$C_｛C｝^（*｝）上的KMS状态集具有丰富的结构。特别地，我们展示了无数类型I、II和III的极端KMS状态。
{"title":"KMS states on \u0000$C_c^{*}(mathbb{N}^2)$","authors":"Anbu Arjunan, Sruthymurali, S. Sundar","doi":"10.1017/S0017089523000071","DOIUrl":"https://doi.org/10.1017/S0017089523000071","url":null,"abstract":"Abstract Let \u0000$C_c^{*}(mathbb{N}^{2})$\u0000 be the universal \u0000$C^{*}$\u0000 -algebra generated by a semigroup of isometries \u0000${v_{(m,n)},:, m,n in mathbb{N}}$\u0000 whose range projections commute. We analyse the structure of KMS states on \u0000$C_{c}^{*}(mathbb{N}^2)$\u0000 for the time evolution determined by a homomorphism \u0000$c,:,mathbb{Z}^{2} to mathbb{R}$\u0000 . In contrast to the reduced version \u0000$C_{red}^{*}(mathbb{N}^{2})$\u0000 , we show that the set of KMS states on \u0000$C_{c}^{*}(mathbb{N}^{2})$\u0000 has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"501 - 528"},"PeriodicalIF":0.5,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44842606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-06DOI: 10.1017/S0017089521000446
A. Usachev
Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.
{"title":"Hausdorff dimension of the set of almost convergent sequences","authors":"A. Usachev","doi":"10.1017/S0017089521000446","DOIUrl":"https://doi.org/10.1017/S0017089521000446","url":null,"abstract":"Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"691 - 697"},"PeriodicalIF":0.5,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43570293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-02DOI: 10.1017/S0017089522000416
G. Bellamy, J. Schmitt, U. Thiel
Abstract Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups. Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.
{"title":"On parabolic subgroups of symplectic reflection groups","authors":"G. Bellamy, J. Schmitt, U. Thiel","doi":"10.1017/S0017089522000416","DOIUrl":"https://doi.org/10.1017/S0017089522000416","url":null,"abstract":"Abstract Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups. Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"401 - 413"},"PeriodicalIF":0.5,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43462559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-02DOI: 10.1017/S0017089521000410
Jesús Rodríguez
Abstract Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.
{"title":"Abelian actions on compact nonorientable Riemann surfaces","authors":"Jesús Rodríguez","doi":"10.1017/S0017089521000410","DOIUrl":"https://doi.org/10.1017/S0017089521000410","url":null,"abstract":"Abstract Given an integer \u0000$g>2$\u0000 , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"634 - 648"},"PeriodicalIF":0.5,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48163514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-21DOI: 10.1017/S0017089522000167
A. Hone
Abstract Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the $mathbb{C}$ -algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.
{"title":"Casting light on shadow Somos sequences","authors":"A. Hone","doi":"10.1017/S0017089522000167","DOIUrl":"https://doi.org/10.1017/S0017089522000167","url":null,"abstract":"Abstract Recently Ovsienko and Tabachnikov considered extensions of Somos and Gale-Robinson sequences, defined over the algebra of dual numbers. Ovsienko used the same idea to construct so-called shadow sequences derived from other nonlinear recurrence relations exhibiting the Laurent phenomenon, with the original motivation being the hope that these examples should lead to an appropriate notion of a cluster superalgebra, incorporating Grassmann variables. Here, we present various explicit expressions for the shadow of Somos-4 sequences and describe the solution of a general Somos-4 recurrence defined over the \u0000$mathbb{C}$\u0000 -algebra of dual numbers from several different viewpoints: analytic formulae in terms of elliptic functions, linear difference equations, and Hankel determinants.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"S87 - S101"},"PeriodicalIF":0.5,"publicationDate":"2021-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42125167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-11DOI: 10.1017/S0017089522000428
Alex Evetts
Abstract We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroups are infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.
{"title":"Conjugacy growth in the higher Heisenberg groups","authors":"Alex Evetts","doi":"10.1017/S0017089522000428","DOIUrl":"https://doi.org/10.1017/S0017089522000428","url":null,"abstract":"Abstract We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroups are infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"S148 - S169"},"PeriodicalIF":0.5,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43119586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-14DOI: 10.1017/S0017089521000331
N. Dummigan
Abstract Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.
{"title":"CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES","authors":"N. Dummigan","doi":"10.1017/S0017089521000331","DOIUrl":"https://doi.org/10.1017/S0017089521000331","url":null,"abstract":"Abstract Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"504 - 525"},"PeriodicalIF":0.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45462562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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