Indagationes Mathematicae-New Series最新文献

{"title":"Dirac cohomology for the BGG category O","authors":"Spyridon Afentoulidis-Almpanis","doi":"10.1016/j.indag.2023.11.001","DOIUrl":"10.1016/j.indag.2023.11.001","url":null,"abstract":"<div><p><span>We study Dirac cohomology </span><span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>h</mi></mrow></msubsup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> for modules belonging to category <span><math><mi>O</mi></math></span><span> of a finite dimensional complex semisimple Lie algebra. We start by studying the generalized infinitesimal character decomposition of </span><span><math><mrow><mi>M</mi><mo>⊗</mo><mi>S</mi></mrow></math></span>, with <span><math><mi>S</mi></math></span> being a spin module of <span><math><msup><mrow><mi>h</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span>. As a consequence, “Vogan’s conjecture” holds, and we prove a nonvanishing result for <span><math><mrow><msubsup><mrow><mi>H</mi></mrow><mrow><mi>D</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>h</mi></mrow></msubsup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow></mrow></math></span> while we show that in the case of a Hermitian symmetric pair <span><math><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> and an irreducible unitary module <span><math><mrow><mi>M</mi><mo>∈</mo><mi>O</mi></mrow></math></span>, Dirac cohomology coincides with the nilpotent Lie algebra cohomology with coefficients in <span><math><mi>M</mi></math></span>. In the last part, we show that the higher Dirac cohomology and index introduced by Pandžić and Somberg satisfy nice homological properties for <span><math><mrow><mi>M</mi><mo>∈</mo><mi>O</mi></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135615809","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}

{"title":"Tensor product of representations of quivers","authors":"Pradeep Das ,&nbsp;Umesh V. Dubey ,&nbsp;N. Raghavendra","doi":"10.1016/j.indag.2024.01.005","DOIUrl":"10.1016/j.indag.2024.01.005","url":null,"abstract":"<div><p>In this article, we define the tensor product <span><math><mrow><mi>V</mi><mo>⊗</mo><mi>W</mi></mrow></math></span> of a representation <span><math><mi>V</mi></math></span> of a quiver <span><math><mi>Q</mi></math></span> with a representation <span><math><mi>W</mi></math></span> of an another quiver <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, and show that the representation <span><math><mrow><mi>V</mi><mo>⊗</mo><mi>W</mi></mrow></math></span> is semistable if <span><math><mi>V</mi></math></span> and <span><math><mi>W</mi></math></span> are semistable. We give a relation between the universal representations on the fine moduli spaces <span><math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> and <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> of representations of <span><math><mrow><mi>Q</mi><mo>,</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span> and <span><math><mrow><mi>Q</mi><mo>⊗</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span><span> respectively over arbitrary algebraically closed fields<span>. We further describe a relation between the natural line bundles on these moduli spaces when the base is the field of complex numbers. We then prove that the internal product </span></span><span><math><mrow><mover><mrow><mi>Q</mi></mrow><mrow><mo>̃</mo></mrow></mover><mo>⊗</mo><mover><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mo>̃</mo></mrow></mover></mrow></math></span> of covering quivers is a sub-quiver of the covering quiver <span><math><mover><mrow><mi>Q</mi><mo>⊗</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><mo>˜</mo></mrow></mover></math></span>. We deduce the relation between stability of the representations <span><math><mover><mrow><mi>V</mi><mo>⊗</mo><mi>W</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and <span><math><mrow><mover><mrow><mi>V</mi></mrow><mrow><mo>̃</mo></mrow></mover><mo>⊗</mo><mover><mrow><mi>W</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow></math></span>, where <span><math><mover><mrow><mi>V</mi></mrow><mrow><mo>̃</mo></mrow></mover></math></span> denotes the lift of the representation <span><math><mi>V</mi></math></span> of <span><math><mi>Q</mi></math></span> to the covering quiver <span><math><mover><mrow><mi>Q</mi></mrow><mrow><mo>̃</mo></mrow></mover></math></span>. We also lift the relation between the natural line bundles on the product of moduli spaces <span><math><mrow><mover><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>̃</mo></mrow></mover><mo>×</mo><mover><mrow><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mo>̃</mo></mrow></mover></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584297","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}

{"title":"Normal forms for principal Poisson Hamiltonian spaces","authors":"Pedro Frejlich ,&nbsp;Ioan Mărcuţ","doi":"10.1016/j.indag.2024.01.001","DOIUrl":"10.1016/j.indag.2024.01.001","url":null,"abstract":"<div><p><span>We prove a normal form theorem for principal </span>Hamiltonian<span><span> actions on Poisson manifolds<span> around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from </span></span>symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map.</span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139413509","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}

{"title":"On the friable mean-value of the Erdős–Hooley Delta function","authors":"B. Martin ,&nbsp;G. Tenenbaum ,&nbsp;J. Wetzer","doi":"10.1016/j.indag.2024.02.002","DOIUrl":"10.1016/j.indag.2024.02.002","url":null,"abstract":"<div><p>For integer <span><math><mi>n</mi></math></span> and real <span><math><mi>u</mi></math></span>, define <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><mrow><mo>|</mo><mrow><mo>{</mo><mi>d</mi><mo>:</mo><mi>d</mi><mo>∣</mo><mi>n</mi><mo>,</mo><mspace></mspace><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi></mrow></msup><mo>&lt;</mo><mi>d</mi><mo>⩽</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>}</mo></mrow><mo>|</mo></mrow></mrow></math></span>. Then, the Erdős–Hooley Delta function is defined as <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>max</mo></mrow><mrow><mi>u</mi><mo>∈</mo><mi>R</mi></mrow></msub><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> We provide uniform upper and lower bounds for the mean-value of <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> over friable integers, i.e. integers free of large prime factors.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000090/pdfft?md5=d2a0f3d37cb93941f7d1335c246fb3a7&pid=1-s2.0-S0019357724000090-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139917661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the transcendence of power towers of Liouville numbers","authors":"Diego Marques ,&nbsp;Marcelo Oliveira ,&nbsp;Pavel Trojovský","doi":"10.1016/j.indag.2023.11.002","DOIUrl":"10.1016/j.indag.2023.11.002","url":null,"abstract":"<div><p>In this paper, among other things, we explicit a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-dense set of Liouville numbers, for which the triple power tower of any of its elements is a transcendental number.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135669305","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}

{"title":"A note on the group extension problem to semi-universal deformation","authors":"An-Khuong Doan","doi":"10.1016/j.indag.2023.11.003","DOIUrl":"10.1016/j.indag.2023.11.003","url":null,"abstract":"<div><p><span>The aim of this note is twofold. Firstly, we explain in detail Remark 4.1 in Doan (2020) by showing that the action of the automorphism group of the second Hirzebruch surface </span><span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span> on itself extends to its formal semi-universal deformation only up to the first order. Secondly, we show that for reductive group actions, the locality of the extended actions on the Kuranishi space constructed in Doan (2021) is the best one could expect in general.</span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138523212","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}

{"title":"Banach function spaces done right","authors":"Emiel Lorist ,&nbsp;Zoe Nieraeth","doi":"10.1016/j.indag.2023.11.004","DOIUrl":"10.1016/j.indag.2023.11.004","url":null,"abstract":"<div><p>In this survey, we discuss the definition of a (quasi-)Banach function space. We advertise the original definition by Zaanen and Luxemburg, which does not have various issues introduced by other, subsequent definitions. Moreover, we prove versions of well-known basic properties of Banach function spaces in the setting of <em>quasi</em>-Banach function spaces.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723001039/pdfft?md5=bbde971ef6c1a863dc397afd75f0a8fb&pid=1-s2.0-S0019357723001039-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138523213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On nth order Euler polynomials of degree n that are Eisenstein","authors":"Michael Filaseta ,&nbsp;Thomas Luckner","doi":"10.1016/j.indag.2023.09.001","DOIUrl":"10.1016/j.indag.2023.09.001","url":null,"abstract":"<div><p>For <span><math><mi>m</mi></math></span> an even positive integer and <span><math><mi>p</mi></math></span> an odd prime, we show that the generalized Euler polynomial <span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>m</mi><mi>p</mi></mrow><mrow><mrow><mo>(</mo><mi>m</mi><mi>p</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> is in Eisenstein form with respect to <span><math><mi>p</mi></math></span> if and only if <span><math><mi>p</mi></math></span> does not divide <span><math><mrow><mi>m</mi><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>. As a consequence, we deduce that at least <span><math><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></math></span> of the generalized Euler polynomials <span><math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> are in Eisenstein form with respect to a prime <span><math><mi>p</mi></math></span> dividing <span><math><mi>n</mi></math></span> and, hence, irreducible over <span><math><mi>Q</mi></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135349570","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}

{"title":"On integral cohomology algebra of some oriented Grassmann manifolds","authors":"Milica Jovanović","doi":"10.1016/j.indag.2023.07.004","DOIUrl":"10.1016/j.indag.2023.07.004","url":null,"abstract":"<div><p><span>The integral cohomology algebra of </span><span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>6</mn><mo>,</mo><mn>3</mn></mrow></msub></math></span> has been determined in the recent work of Kalafat and Yalçınkaya. We completely determine the integral cohomology algebra of <span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi><mo>,</mo><mn>3</mn></mrow></msub></math></span> for <span><math><mrow><mi>n</mi><mo>=</mo><mn>8</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>=</mo><mn>10</mn></mrow></math></span><span>. The main method used to describe these algebras is the Leray–Serre spectral sequence. We also illustrate this method by determining the integral cohomology algebra of </span><span><math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>n</mi><mo>,</mo><mn>2</mn></mrow></msub></math></span> for <span><math><mi>n</mi></math></span> odd.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48696144","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}

{"title":"The Mathieu conjecture for SU(2) reduced to an abelian conjecture","authors":"Michael Müger ,&nbsp;Lars Tuset","doi":"10.1016/j.indag.2023.10.001","DOIUrl":"10.1016/j.indag.2023.10.001","url":null,"abstract":"<div><p>We reduce the Mathieu conjecture for <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> to a conjecture about moments of Laurent polynomials in two variables with single variable polynomial coefficients.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000939/pdfft?md5=cf524c22f110e1e5fbeb63f0f4fe9fe5&pid=1-s2.0-S0019357723000939-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135922117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}