Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110084
Noy Soffer Aranov , Taehyeong Kim
We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of ε-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier (2016) [9].
{"title":"Hausdorff dimension of singular vectors in function fields","authors":"Noy Soffer Aranov , Taehyeong Kim","doi":"10.1016/j.aim.2024.110084","DOIUrl":"10.1016/j.aim.2024.110084","url":null,"abstract":"<div><div>We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of <em>ε</em>-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier (2016) <span><span>[9]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110084"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164294","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 : 2025-02-01DOI: 10.1016/j.aim.2024.110054
Mihalis Mourgoglou , Thanasis Zacharopoulos
<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, be an open set with <em>s</em>-Ahlfors regular boundary ∂Ω, for some <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span>, such that either <span><math><mi>s</mi><mo>=</mo><mi>n</mi></math></span> and Ω is a corkscrew domain with the pointwise John condition, or <span><math><mi>s</mi><mo><</mo><mi>n</mi></math></span> and <span><math><mi>Ω</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>E</mi></math></span>, for some <em>s</em>-Ahlfors regular set <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. In this paper we provide a unifying method to construct Varopoulos type extensions of BMO and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> boundary functions. In particular, we show that a) if <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BMO</mi></mrow><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, there exists <span><math><mi>F</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that <span><math><mtext>dist</mtext><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> is uniformly bounded in Ω and the Carleson functional of <span><math><mtext>dist</mtext><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> as well the sharp non-tangential maximal function of <em>F</em> are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of <em>f</em>, and <span><math><mi>F</mi><mo>→</mo><mi>f</mi></math></span> in a certain non-tangential sense <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mo>|</mo></mrow><mrow><mo>∂</mo><mi>Ω</mi></mrow></msub></math></span>-almost everywhere; b) if <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>, there exists <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that the non-tangential maximal functions of <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> and <span><math><
{"title":"Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with L∞ coefficients","authors":"Mihalis Mourgoglou , Thanasis Zacharopoulos","doi":"10.1016/j.aim.2024.110054","DOIUrl":"10.1016/j.aim.2024.110054","url":null,"abstract":"<div><div>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, be an open set with <em>s</em>-Ahlfors regular boundary ∂Ω, for some <span><math><mi>s</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span>, such that either <span><math><mi>s</mi><mo>=</mo><mi>n</mi></math></span> and Ω is a corkscrew domain with the pointwise John condition, or <span><math><mi>s</mi><mo><</mo><mi>n</mi></math></span> and <span><math><mi>Ω</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∖</mo><mi>E</mi></math></span>, for some <em>s</em>-Ahlfors regular set <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. In this paper we provide a unifying method to construct Varopoulos type extensions of BMO and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> boundary functions. In particular, we show that a) if <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BMO</mi></mrow><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, there exists <span><math><mi>F</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that <span><math><mtext>dist</mtext><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> is uniformly bounded in Ω and the Carleson functional of <span><math><mtext>dist</mtext><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi><mo>−</mo><mi>n</mi></mrow></msup><mo>|</mo><mi>∇</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math></span> as well the sharp non-tangential maximal function of <em>F</em> are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of <em>f</em>, and <span><math><mi>F</mi><mo>→</mo><mi>f</mi></math></span> in a certain non-tangential sense <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><msub><mrow><mo>|</mo></mrow><mrow><mo>∂</mo><mi>Ω</mi></mrow></msub></math></span>-almost everywhere; b) if <span><math><mover><mrow><mi>f</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span>, <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span>, there exists <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> such that the non-tangential maximal functions of <span><math><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> and <span><math><","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110054"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164296","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110042
Kirill Lazebnik
We characterize which planar graphs arise as the pullback, under a rational map r, of an analytic Jordan curve passing through the critical values of r. We also prove that such pullbacks are dense within the collection of , where f is a branched cover of the sphere and Σ is a Jordan curve passing through the branched values of f.
{"title":"Analytic and topological nets","authors":"Kirill Lazebnik","doi":"10.1016/j.aim.2024.110042","DOIUrl":"10.1016/j.aim.2024.110042","url":null,"abstract":"<div><div>We characterize which planar graphs arise as the pullback, under a rational map <em>r</em>, of an analytic Jordan curve passing through the critical values of <em>r</em>. We also prove that such pullbacks are dense within the collection of <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>Σ</mi><mo>)</mo></math></span>, where <em>f</em> is a branched cover of the sphere and Σ is a Jordan curve passing through the branched values of <em>f</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110042"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164290","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 : 2025-02-01DOI: 10.1016/j.aim.2024.110024
Francesco Bonsante, Michael Wolf
We prove that the space of circle packings compatible with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.
{"title":"Projective rigidity of circle packings","authors":"Francesco Bonsante, Michael Wolf","doi":"10.1016/j.aim.2024.110024","DOIUrl":"10.1016/j.aim.2024.110024","url":null,"abstract":"<div><div>We prove that the space of circle packings compatible with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110024"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164299","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110083
Haowu Wang , Brandon Williams
In this paper we consider several problems in the theory of automorphic products and generalized Kac–Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type and certain tuples of Jacobi forms of level N.
{"title":"The fake monster algebra and singular Borcherds products","authors":"Haowu Wang , Brandon Williams","doi":"10.1016/j.aim.2024.110083","DOIUrl":"10.1016/j.aim.2024.110083","url":null,"abstract":"<div><div>In this paper we consider several problems in the theory of automorphic products and generalized Kac–Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type <span><math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>⊕</mo><mi>U</mi><mo>⊕</mo><mi>L</mi></math></span> and certain tuples of Jacobi forms of level <em>N</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110083"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164293","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 : 2025-02-01DOI: 10.1016/j.aim.2024.110098
Xilun Li , Shengxuan Zhou
For any integers , we construct a Ricci limit space such that for a fixed point, some tangent cones are and some are . This is an improvement of Menguy's example [3]. Moreover, we show that for any finite collection of closed Riemannian manifolds with , there exists a collapsed Ricci limit space such that each Riemannian cone is a tangent cone of X at x.
{"title":"New exotic examples of Ricci limit spaces","authors":"Xilun Li , Shengxuan Zhou","doi":"10.1016/j.aim.2024.110098","DOIUrl":"10.1016/j.aim.2024.110098","url":null,"abstract":"<div><div>For any integers <span><math><mi>m</mi><mo>⩾</mo><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>, we construct a Ricci limit space <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> such that for a fixed point, some tangent cones are <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and some are <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. This is an improvement of Menguy's example <span><span>[3]</span></span>. Moreover, we show that for any finite collection of closed Riemannian manifolds <span><math><mo>(</mo><msubsup><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msubsup><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> with <span><math><msub><mrow><mi>Ric</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>⩾</mo><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><mn>1</mn><mo>)</mo><mo>⩾</mo><mn>1</mn></math></span>, there exists a collapsed Ricci limit space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> such that each Riemannian cone <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is a tangent cone of <em>X</em> at <em>x</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110098"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150241","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 : 2025-02-01DOI: 10.1016/j.aim.2024.110092
Dennis Borisov , Ludmil Katzarkov , Artan Sheshmani
It is proved that derived -schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part of a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi–Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.
{"title":"Shifted symplectic structures on derived Quot-stacks II – derived Quot-schemes as dg manifolds","authors":"Dennis Borisov , Ludmil Katzarkov , Artan Sheshmani","doi":"10.1016/j.aim.2024.110092","DOIUrl":"10.1016/j.aim.2024.110092","url":null,"abstract":"<div><div>It is proved that derived <span><math><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi></math></span>-schemes, as defined by Ciocan-Fontanine and Kapranov, are represented by dg manifolds of finite type. This is the second part of a work aimed to analyze shifted symplectic structures on moduli spaces of coherent sheaves on Calabi–Yau manifolds. The first part related dg manifolds to derived schemes as defined by Toën and Vezzosi.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110092"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150249","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110101
Tao Su
The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension d over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension . Via a microlocal geometric perspective, we verify this conjecture for a class of rank n wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an n-strand positive braid.
{"title":"Dual boundary complexes of Betti moduli spaces over the two-sphere with one irregular singularity","authors":"Tao Su","doi":"10.1016/j.aim.2024.110101","DOIUrl":"10.1016/j.aim.2024.110101","url":null,"abstract":"<div><div>The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C. Simpson states that, a smooth Betti moduli space of complex dimension <em>d</em> over a punctured Riemann surface has the dual boundary complex homotopy equivalent to a sphere of dimension <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span>. Via a microlocal geometric perspective, we verify this conjecture for a class of rank <em>n</em> wild character varieties over the two-sphere with one puncture, associated with any Stokes Legendrian link defined by an <em>n</em>-strand positive braid.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110101"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150250","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 : 2025-02-01DOI: 10.1016/j.aim.2024.110090
Chris Bowman , Rowena Paget , Mark Wildon
The plethysm coefficient is the multiplicity of the Schur function in the plethysm product . In this paper we use Schur–Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of ν, μ and λ are all large. In particular, it gives the first positive formula in the case when ν and λ are arbitrary and μ has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients , and when m and n are large.
{"title":"The partition algebra and the plethysm coefficients II: Ramified plethysm","authors":"Chris Bowman , Rowena Paget , Mark Wildon","doi":"10.1016/j.aim.2024.110090","DOIUrl":"10.1016/j.aim.2024.110090","url":null,"abstract":"<div><div>The plethysm coefficient <span><math><mi>p</mi><mo>(</mo><mi>ν</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> is the multiplicity of the Schur function <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> in the plethysm product <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>∘</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span>. In this paper we use Schur–Weyl duality between wreath products of symmetric groups and the ramified partition algebra to interpret an arbitrary plethysm coefficient as the multiplicity of an appropriate composition factor in the restriction of a module for the ramified partition algebra to the partition algebra. This result implies new stability phenomenon for plethysm coefficients when the first parts of <em>ν</em>, <em>μ</em> and <em>λ</em> are all large. In particular, it gives the first positive formula in the case when <em>ν</em> and <em>λ</em> are arbitrary and <em>μ</em> has one part. Corollaries include new explicit positive formulae and combinatorial interpretations for the plethysm coefficients <span><math><mi>p</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>)</mo></math></span>, and <span><math><mi>p</mi><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>b</mi><mo>,</mo><msup><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></msup><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>m</mi><mi>n</mi><mo>−</mo><mi>r</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>)</mo></math></span> when <em>m</em> and <em>n</em> are large.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110090"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149611","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110093
Théo Pinet
<div><div>Fix a finite-dimensional simple Lie algebra <span><math><mi>g</mi></math></span> and let <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>⊆</mo><mi>g</mi></math></span> be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span>–modules.</div><div>In this article, we study Finkelberg–Tsymbaliuk's shifted quantum affine algebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mo>(</mo><mi>g</mi><mo>)</mo></math></span> and the associated categories <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>μ</mi></mrow></msup></math></span> (defined by Hernandez). In particular, we introduce natural subalgebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo><mo>⊆</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mo>(</mo><mi>g</mi><mo>)</mo></math></span> and obtain a functor <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> from <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msup><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>μ</mi></mrow></msub><msup><mrow><mi>O</mi></mrow><mrow><mi>μ</mi></mrow></msup></math></span> to <span><math><msub><mrow><mo>⨁</mo></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo><mtext>-Mod</mtext><mo>)</mo></math></span> using the canonical restriction functors. We then establish that <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call <em>inflations</em>.</div><div>We conjecture that all simple objects in <span><math><msubsup><mrow><mi>O</mi></mrow><mrow><mi>J</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msubsup></math></span> (which is the analog of <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msup></math></span> for the subalgebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo></math></span>) admit some inflation and prove this for <span><math><mi>g</mi></math></span> of type A–B or <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> a direct sum of copies of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub
{"title":"Inflations for representations of shifted quantum affine algebras","authors":"Théo Pinet","doi":"10.1016/j.aim.2024.110093","DOIUrl":"10.1016/j.aim.2024.110093","url":null,"abstract":"<div><div>Fix a finite-dimensional simple Lie algebra <span><math><mi>g</mi></math></span> and let <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>⊆</mo><mi>g</mi></math></span> be a Lie subalgebra coming from a Dynkin diagram inclusion. Then, the corresponding restriction functor is not essentially surjective on finite-dimensional simple <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span>–modules.</div><div>In this article, we study Finkelberg–Tsymbaliuk's shifted quantum affine algebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mo>(</mo><mi>g</mi><mo>)</mo></math></span> and the associated categories <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>μ</mi></mrow></msup></math></span> (defined by Hernandez). In particular, we introduce natural subalgebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo><mo>⊆</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>μ</mi></mrow></msubsup><mo>(</mo><mi>g</mi><mo>)</mo></math></span> and obtain a functor <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> from <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msup><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>μ</mi></mrow></msub><msup><mrow><mi>O</mi></mrow><mrow><mi>μ</mi></mrow></msup></math></span> to <span><math><msub><mrow><mo>⨁</mo></mrow><mrow><mi>ν</mi></mrow></msub><mo>(</mo><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo><mtext>-Mod</mtext><mo>)</mo></math></span> using the canonical restriction functors. We then establish that <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> is essentially surjective on finite-dimensional simple objects by constructing notable preimages that we call <em>inflations</em>.</div><div>We conjecture that all simple objects in <span><math><msubsup><mrow><mi>O</mi></mrow><mrow><mi>J</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msubsup></math></span> (which is the analog of <span><math><msup><mrow><mi>O</mi></mrow><mrow><mi>s</mi><mi>h</mi></mrow></msup></math></span> for the subalgebras <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>ν</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub><mo>)</mo></math></span>) admit some inflation and prove this for <span><math><mi>g</mi></math></span> of type A–B or <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>J</mi></mrow></msub></math></span> a direct sum of copies of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110093"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149614","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}
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