Pub Date : 2021-06-01DOI: 10.1515/crelle-2021-frontmatter775
{"title":"Frontmatter","authors":"","doi":"10.1515/crelle-2021-frontmatter775","DOIUrl":"https://doi.org/10.1515/crelle-2021-frontmatter775","url":null,"abstract":"","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85121649","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}
Pub Date : 2021-05-27DOI: 10.1515/crelle-2021-0048
Jianya Liu, Lilu Zhao
Abstract It is proved that all sufficiently large integers n can be represented as n=x12+x23+⋯+x1314,n=x_{1}^{2}+x_{2}^{3}+cdots+x_{13}^{14}, where x1,…,x13{x_{1},ldots,x_{13}} are positive integers. This improves upon the current record with fourteen variables in place of thirteen.
{"title":"Representation by sums of unlike powers","authors":"Jianya Liu, Lilu Zhao","doi":"10.1515/crelle-2021-0048","DOIUrl":"https://doi.org/10.1515/crelle-2021-0048","url":null,"abstract":"Abstract It is proved that all sufficiently large integers n can be represented as n=x12+x23+⋯+x1314,n=x_{1}^{2}+x_{2}^{3}+cdots+x_{13}^{14}, where x1,…,x13{x_{1},ldots,x_{13}} are positive integers. This improves upon the current record with fourteen variables in place of thirteen.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86665276","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}
Pub Date : 2021-05-26DOI: 10.1515/crelle-2023-0050
Hansol Hong, Artur B. Saturnino
Abstract In this paper, we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of R 3 mathbb{R}^{3} which is minimal or has a contact angle less than or equal to π / 2 pi/2 must be a half-plane. Using this uniqueness result, we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.
{"title":"Capillary surfaces: Stability, index and curvature estimates","authors":"Hansol Hong, Artur B. Saturnino","doi":"10.1515/crelle-2023-0050","DOIUrl":"https://doi.org/10.1515/crelle-2023-0050","url":null,"abstract":"Abstract In this paper, we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of R 3 mathbb{R}^{3} which is minimal or has a contact angle less than or equal to π / 2 pi/2 must be a half-plane. Using this uniqueness result, we obtain curvature estimates for strongly stable capillary surfaces immersed in a 3-manifold with bounded geometry.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90234181","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}
Pub Date : 2021-05-22DOI: 10.1515/crelle-2023-0042
Xu Cheng, Detang Zhou
Abstract Let ( M , g , f ) {{(M,g,f)}} be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric + ∇ 2 f = λ g {{mathrm{Ric}+nabla^{2}f=lambda g}} , where λ {{lambda}} is a positive real number. We prove that if M {{M}} has constant scalar curvature S = 2 λ {{S=2lambda}} , it must be a quotient of 𝕊 2 × ℝ 2 {{mathbb{S}^{2}timesmathbb{R}^{2}}} . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ 4 {{mathbb{R}^{4}}} , 𝕊 2 × ℝ 2 {{mathbb{S}^{2}timesmathbb{R}^{2}}} or 𝕊 3 × ℝ {{mathbb{S}^{3}timesmathbb{R}}} .
{"title":"Rigidity of four-dimensional gradient shrinking Ricci solitons","authors":"Xu Cheng, Detang Zhou","doi":"10.1515/crelle-2023-0042","DOIUrl":"https://doi.org/10.1515/crelle-2023-0042","url":null,"abstract":"Abstract Let ( M , g , f ) {{(M,g,f)}} be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric + ∇ 2 f = λ g {{mathrm{Ric}+nabla^{2}f=lambda g}} , where λ {{lambda}} is a positive real number. We prove that if M {{M}} has constant scalar curvature S = 2 λ {{S=2lambda}} , it must be a quotient of 𝕊 2 × ℝ 2 {{mathbb{S}^{2}timesmathbb{R}^{2}}} . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ 4 {{mathbb{R}^{4}}} , 𝕊 2 × ℝ 2 {{mathbb{S}^{2}timesmathbb{R}^{2}}} or 𝕊 3 × ℝ {{mathbb{S}^{3}timesmathbb{R}}} .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82185703","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}
Pub Date : 2021-05-18DOI: 10.1515/crelle-2021-0022
J. Schwermer
Abstract The cohomology H*(Γ,E){H^{*}(Gamma,E)} of a torsion-free arithmetic subgroup Γ of the special linear ℚ{mathbb{Q}}-group 𝖦=SLn{mathsf{G}={mathrm{SL}}_{n}} may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes {𝖯}{{mathsf{P}}} of associate proper parabolic ℚ{mathbb{Q}}-subgroups of 𝖦{mathsf{G}}. Each summand H{P}*(Γ,E){H^{*}_{mathrm{{P}}}(Gamma,E)} is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in {𝖯}{{mathsf{P}}}. The cohomology H*(Γ,E){H^{*}(Gamma,E)} vanishes above the degree given by the cohomological dimension cd(Γ)=12n(n-1){mathrm{cd}(Gamma)=frac{1}{2}n(n-1)}. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {𝖯}{{mathsf{P}}} for which the corresponding summand H{𝖯}cd(Γ)(Γ,E){H^{mathrm{cd}(Gamma)}_{mathrm{{mathsf{P}}}}(Gamma,E)} vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H{𝖰}cd(Γ)(Γ,ℂ){H^{mathrm{cd}(Gamma)}_{mathrm{{mathsf{Q}}}}(Gamma,mathbb{C})}. Finally, in the case of a principal congruence subgroup Γ(q){Gamma(q)}, q=pν>5{q=p^{nu}>5}, p≥3{pgeq 3} a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes {𝖰}{{mathsf{Q}}}, there is a precise formula for their dimension.
{"title":"Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ","authors":"J. Schwermer","doi":"10.1515/crelle-2021-0022","DOIUrl":"https://doi.org/10.1515/crelle-2021-0022","url":null,"abstract":"Abstract The cohomology H*(Γ,E){H^{*}(Gamma,E)} of a torsion-free arithmetic subgroup Γ of the special linear ℚ{mathbb{Q}}-group 𝖦=SLn{mathsf{G}={mathrm{SL}}_{n}} may be interpreted in terms of the automorphic spectrum of Γ. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes {𝖯}{{mathsf{P}}} of associate proper parabolic ℚ{mathbb{Q}}-subgroups of 𝖦{mathsf{G}}. Each summand H{P}*(Γ,E){H^{*}_{mathrm{{P}}}(Gamma,E)} is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in {𝖯}{{mathsf{P}}}. The cohomology H*(Γ,E){H^{*}(Gamma,E)} vanishes above the degree given by the cohomological dimension cd(Γ)=12n(n-1){mathrm{cd}(Gamma)=frac{1}{2}n(n-1)}. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {𝖯}{{mathsf{P}}} for which the corresponding summand H{𝖯}cd(Γ)(Γ,E){H^{mathrm{cd}(Gamma)}_{mathrm{{mathsf{P}}}}(Gamma,E)} vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span H{𝖰}cd(Γ)(Γ,ℂ){H^{mathrm{cd}(Gamma)}_{mathrm{{mathsf{Q}}}}(Gamma,mathbb{C})}. Finally, in the case of a principal congruence subgroup Γ(q){Gamma(q)}, q=pν>5{q=p^{nu}>5}, p≥3{pgeq 3} a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes {𝖰}{{mathsf{Q}}}, there is a precise formula for their dimension.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91173569","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}
Pub Date : 2021-05-12DOI: 10.1515/crelle-2021-0019
C. McMullen
Abstract This paper introduces a space of nonabelian modular symbols 𝒮(V){{mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ωω+1{omega^{omega}+1}.
{"title":"Modular symbols for Teichmüller curves","authors":"C. McMullen","doi":"10.1515/crelle-2021-0019","DOIUrl":"https://doi.org/10.1515/crelle-2021-0019","url":null,"abstract":"Abstract This paper introduces a space of nonabelian modular symbols 𝒮(V){{mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ωω+1{omega^{omega}+1}.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82267404","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}
Pub Date : 2021-05-11DOI: 10.1515/crelle-2023-0013
Piotr Achinger, Marcin Lara, Alex Youcis
Abstract We develop the notion of a geometric covering of a rigid space 𝑋, which yields a larger class of covering spaces than that studied previously by de Jong. Geometric coverings are closed under disjoint unions and are étale local on 𝑋. If 𝑋 is connected, its geometric coverings form a tame infinite Galois category and hence are classified by a topological group. The definition is based on the property of lifting of “geometric arcs” and is meant to be an analogue of the notion developed for schemes by Bhatt and Scholze.
{"title":"Geometric arcs and fundamental groups of rigid spaces","authors":"Piotr Achinger, Marcin Lara, Alex Youcis","doi":"10.1515/crelle-2023-0013","DOIUrl":"https://doi.org/10.1515/crelle-2023-0013","url":null,"abstract":"Abstract We develop the notion of a geometric covering of a rigid space 𝑋, which yields a larger class of covering spaces than that studied previously by de Jong. Geometric coverings are closed under disjoint unions and are étale local on 𝑋. If 𝑋 is connected, its geometric coverings form a tame infinite Galois category and hence are classified by a topological group. The definition is based on the property of lifting of “geometric arcs” and is meant to be an analogue of the notion developed for schemes by Bhatt and Scholze.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76467722","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}
Pub Date : 2021-05-10DOI: 10.1515/crelle-2022-0026
Sergio Cruz-Blázquez, A. Malchiodi, D. Ruiz
Abstract We consider the case with boundary of the classical Kazdan–Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.
{"title":"Conformal metrics with prescribed scalar and mean curvature","authors":"Sergio Cruz-Blázquez, A. Malchiodi, D. Ruiz","doi":"10.1515/crelle-2022-0026","DOIUrl":"https://doi.org/10.1515/crelle-2022-0026","url":null,"abstract":"Abstract We consider the case with boundary of the classical Kazdan–Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85839319","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}
Pub Date : 2021-04-28DOI: 10.1515/crelle-2023-0004
M. Kapovich, Alex Kontorovich
Abstract We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ {{mathbb{Q}}} -arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π m {frac{pi}{m}} for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.
摘要:我们提出了Kleinian球填充的概念，这是在[a]中定义的“晶体”(类阿波罗)球填充的推广。孔托洛维奇和K. Nakamura，晶体球填充的几何和算法，国立国立大学学报(自然科学版)。学术科学中国生物医学工程学报，2019,24(2):436-441。与晶体充填不同，Kleinian充填存在于所有维度，“超积分”也是如此。我们将算术定理推广到Kleinian包，即超积分包来自于最简型的π -算术格{{mathbb{Q}}}。这同样适用于更一般的物体，我们称之为Kleinian Bugs，其中球体不必是不相交的，但对于有限多个m，可以满足二面角π m {frac{pi}{m}}。我们解决了Kontorovich和Nakamura(2019)提出的两个问题:(i)算术定理在数域上一般是错误的，(ii)积分填充只产生于非均匀格。
{"title":"On superintegral Kleinian sphere packings, bugs, and arithmetic groups","authors":"M. Kapovich, Alex Kontorovich","doi":"10.1515/crelle-2023-0004","DOIUrl":"https://doi.org/10.1515/crelle-2023-0004","url":null,"abstract":"Abstract We develop the notion of a Kleinian Sphere Packing, a generalization of “crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura, Geometry and arithmetic of crystallographic sphere packings, Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ {{mathbb{Q}}} -arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π m {frac{pi}{m}} for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90627683","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}
Pub Date : 2021-04-26DOI: 10.1515/crelle-2022-0080
M. Gorelik, T. Heidersdorf
Abstract We establish an explicit formula for the character of an irreducible finite-dimensional representation of g l ( m | n ) mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ mathcal{E}_{mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ mathcal{E}_{mu} in the g l ( m | n ) mathfrak{gl}(m|n) and o s p ( m | 2 n ) mathfrak{osp}(m|2n) -case under the map ds mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 mathfrak{g}_{0} -decompositions for g l ( m | n ) mathfrak{gl}(m|n) and o s p ( m | 2 n ) mathfrak{osp}(m|2n) .
摘要建立了有限维不可约表示g¹¹(m|n) mathfrak{gl}(m|n)的一个显式表达式。该公式是基于字符环的基E μ mathcal{E}_{mu}(欧拉字符)的整数系数有限和。我们证明了由Duflo-Serganova上同调函子ds mathrm{ds}导出的超字符环上映射ds mathrm{ds}下，E μ mathcal{E}_{mu}在g _ l _ (m|n) mathfrak{gl}(m|n)和o _ s _ p _ (m|2 _ n) mathfrak{osp}(m|2n) -情况下的“逆”行为的一个简单公式。作为应用，我们得到了超维数、维数和g 0 mathfrak{g}_{0}的组合公式——分解为g _1 (m|n) mathfrak{gl}(m|n)和0 _ s _ p _ (m|2n) mathfrak{osp}(m|2n)。
{"title":"Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor","authors":"M. Gorelik, T. Heidersdorf","doi":"10.1515/crelle-2022-0080","DOIUrl":"https://doi.org/10.1515/crelle-2022-0080","url":null,"abstract":"Abstract We establish an explicit formula for the character of an irreducible finite-dimensional representation of g l ( m | n ) mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ mathcal{E}_{mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ mathcal{E}_{mu} in the g l ( m | n ) mathfrak{gl}(m|n) and o s p ( m | 2 n ) mathfrak{osp}(m|2n) -case under the map ds mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 mathfrak{g}_{0} -decompositions for g l ( m | n ) mathfrak{gl}(m|n) and o s p ( m | 2 n ) mathfrak{osp}(m|2n) .","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78142415","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}
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