Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.08.003
Claudio Pedrini
Let be a smooth cubic fourfold. A well known conjecture asserts that is rational if and only if there a Hodge theoretically associated K3 surface . The surface can be associated to in two other different ways. If there is an equivalence of categories where is the Kuznetsov component of and is a Brauer class, or if there is an isomorphism between the transcendental motive and the (twisted ) transcendental motive of a K3 surface . In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.
{"title":"K3 surfaces associated to a cubic fourfold","authors":"Claudio Pedrini","doi":"10.1016/j.indag.2024.08.003","DOIUrl":"10.1016/j.indag.2024.08.003","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>X</mi><mo>⊂</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> be a smooth cubic fourfold. A well known conjecture asserts that <span><math><mi>X</mi></math></span> is rational if and only if there a Hodge theoretically associated K3 surface <span><math><mi>S</mi></math></span>. The surface <span><math><mi>S</mi></math></span> can be associated to <span><math><mi>X</mi></math></span> in two other different ways. If there is an equivalence of categories <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>≃</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>b</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> where <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is the Kuznetsov component of <span><math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mi>b</mi></mrow></msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>α</mi></math></span> is a Brauer class, or if there is an isomorphism between the transcendental motive <span><math><mrow><mi>t</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and the (twisted ) transcendental motive of a K3 surface <span><math><mi>S</mi></math></span>. In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 158-177"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142248891","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 : 2026-01-01DOI: 10.1016/j.indag.2024.11.002
Rob de Jeu
Let be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.
{"title":"Bounding generators for the kernel and cokernel of the tame symbol for curves","authors":"Rob de Jeu","doi":"10.1016/j.indag.2024.11.002","DOIUrl":"10.1016/j.indag.2024.11.002","url":null,"abstract":"<div><div>Let <span><math><mi>C</mi></math></span> be a regular, irreducible curve that is projective over a field. We obtain bounds in terms of the arithmetic genus of <span><math><mi>C</mi></math></span> for the generators that are required for the cokernel of the tame symbol, as well as, under a simplifying assumption, its kernel. We briefly discuss a potential application to Chow groups.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 281-293"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898100","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 : 2026-01-01DOI: 10.1016/j.indag.2024.05.009
Bruno Kahn
We discuss cases where Voevodsky’s smash nilpotence conjecture is known, and give a few new ones. In particular we explain a theorem of the cube for 1-cycles, which is due to Oussama Ouriachi.
{"title":"Some remarks on the smash-nilpotence conjecture","authors":"Bruno Kahn","doi":"10.1016/j.indag.2024.05.009","DOIUrl":"10.1016/j.indag.2024.05.009","url":null,"abstract":"<div><div>We discuss cases where Voevodsky’s smash nilpotence conjecture is known, and give a few new ones. In particular we explain a theorem of the cube for 1-cycles, which is due to Oussama Ouriachi.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 137-148"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553293","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 : 2026-01-01DOI: 10.1016/j.indag.2024.11.005
Marc Levine
We prove an analog of the virtual localization theorem of Graber–Pandharipande, in the setting of an action by the normalizer of the torus in , and with the Chow groups replaced by the cohomology of a suitably twisted sheaf of Witt groups.
{"title":"Virtual localization in equivariant Witt cohomology","authors":"Marc Levine","doi":"10.1016/j.indag.2024.11.005","DOIUrl":"10.1016/j.indag.2024.11.005","url":null,"abstract":"<div><div>We prove an analog of the virtual localization theorem of Graber–Pandharipande, in the setting of an action by the normalizer of the torus in <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, and with the Chow groups replaced by the cohomology of a suitably twisted sheaf of Witt groups.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 303-375"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898102","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 : 2026-01-01DOI: 10.1016/j.indag.2024.09.009
Joseph Ayoub
In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.
{"title":"Weil cohomology theories and their motivic Hopf algebroids","authors":"Joseph Ayoub","doi":"10.1016/j.indag.2024.09.009","DOIUrl":"10.1016/j.indag.2024.09.009","url":null,"abstract":"<div><div>In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 195-249"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898097","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 : 2025-07-08DOI: 10.1016/j.indag.2025.05.011
Pierre-Antoine Bernard , Étienne Poliquin , Luc Vinet
Two -analogs of the hypercube graph are introduced and shown to be related through a graph quotient. The roles of the subspace lattice graph, of a twisted primitive element of and of the dual -Krawtchouk polynomials are elaborated upon. This paper is dedicated to Tom Koornwinder.
{"title":"A tale of two q-deformations: Connecting dual polar graphs and weighted hypercubes","authors":"Pierre-Antoine Bernard , Étienne Poliquin , Luc Vinet","doi":"10.1016/j.indag.2025.05.011","DOIUrl":"10.1016/j.indag.2025.05.011","url":null,"abstract":"<div><div>Two <span><math><mi>q</mi></math></span>-analogs of the hypercube graph are introduced and shown to be related through a graph quotient. The roles of the subspace lattice graph, of a twisted primitive element of <span><math><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>su</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> and of the dual <span><math><mi>q</mi></math></span>-Krawtchouk polynomials are elaborated upon. This paper is dedicated to Tom Koornwinder.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1779-1794"},"PeriodicalIF":0.8,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374399","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 : 2025-06-30DOI: 10.1016/j.indag.2025.05.013
Yuanyuan Xie
For self-similar measures with overlaps, closed formulas of the -spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. 106 (2019), 56–103]. We extend the results of Ngai and the author (Ngai and Xie, 2019) to graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the -spectrum has been studied by Edgar and Mauldin (1992). The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures on (), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the -spectrum of for , and prove the differentiability of the -spectrum. The main ingredients of this framework include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimensions.
{"title":"Lq-spectrum of graph-directed self-similar measures that have overlaps and are essentially of finite type","authors":"Yuanyuan Xie","doi":"10.1016/j.indag.2025.05.013","DOIUrl":"10.1016/j.indag.2025.05.013","url":null,"abstract":"<div><div>For self-similar measures with overlaps, closed formulas of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. <strong>106</strong> (2019), 56–103]. We extend the results of Ngai and the author (Ngai and Xie, 2019) to graph-directed self-similar measures. For graph-directed self-similar measures satisfying the graph open set condition, the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-spectrum has been studied by Edgar and Mauldin (1992). The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition. For graph-directed self-similar measures <span><math><mi>μ</mi></math></span> on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>\u0000 (<span><math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math></span>), which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-spectrum of <span><math><mi>μ</mi></math></span> for <span><math><mrow><mi>q</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, and prove the differentiability of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math></span>-spectrum. The main ingredients of this framework include graph-directed self-similar measures that are strongly connected and not strongly connected and those in higher dimensions.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1355-1404"},"PeriodicalIF":0.8,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895928","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 : 2025-06-28DOI: 10.1016/j.indag.2025.05.014
Oualid Bouabdillah
<div><div>In a finite subset <span><math><mrow><mi>E</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> of the torus <span><math><mrow><mi>T</mi><mo>=</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, the notion of Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in Bouabdillah and Le Merdy (2024).</div><div>In this paper, we define a quadratic functional calculus for a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>, by a decomposition of type Franks–McIntosh. We show that with some hypothesis on the cotype of <span><math><mi>X</mi></math></span>, this notion is equivalent to the existence of a bounded functional calculus on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</div><div>We define for a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator on a Banach space <span><math><mi>X</mi></math></span> and for any positive real number <span><math><mi>α</mi></math></span> and for any <span><math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math></span> <span><math><mrow><msub><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow><mrow><mi>T</mi><mo>,</mo><mi>α</mi></mrow></msub><mo>=</mo><munder><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></munder><mrow><mo>‖</mo></mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><msup><mrow><mrow><mo>(</mo><mi>I</mi><mo>−</mo><mover><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>¯</mo></mover><mi>T</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>‖</mo></mrow></mrow><mrow><mi>Rad</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></msub><mo>.</mo></mrow></math></span> We show that, under the condition of finite cotype of <span><math><mi>X</mi></math></span>, a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator admits a quadratic functional calculus if and only if the estimates <span><math><mrow><msub><mrow><mo>‖</mo><
{"title":"Square functions associated with RittE operators","authors":"Oualid Bouabdillah","doi":"10.1016/j.indag.2025.05.014","DOIUrl":"10.1016/j.indag.2025.05.014","url":null,"abstract":"<div><div>In a finite subset <span><math><mrow><mi>E</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> of the torus <span><math><mrow><mi>T</mi><mo>=</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span>, the notion of Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in Bouabdillah and Le Merdy (2024).</div><div>In this paper, we define a quadratic functional calculus for a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>, by a decomposition of type Franks–McIntosh. We show that with some hypothesis on the cotype of <span><math><mi>X</mi></math></span>, this notion is equivalent to the existence of a bounded functional calculus on <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</div><div>We define for a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator on a Banach space <span><math><mi>X</mi></math></span> and for any positive real number <span><math><mi>α</mi></math></span> and for any <span><math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math></span>\u0000 <span><math><mrow><msub><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow><mrow><mi>T</mi><mo>,</mo><mi>α</mi></mrow></msub><mo>=</mo><munder><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></munder><mrow><mo>‖</mo></mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mrow><mi>k</mi></mrow><mrow><mi>α</mi><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><munderover><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><msup><mrow><mrow><mo>(</mo><mi>I</mi><mo>−</mo><mover><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mo>¯</mo></mover><mi>T</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>‖</mo></mrow></mrow><mrow><mi>Rad</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></msub><mo>.</mo></mrow></math></span> We show that, under the condition of finite cotype of <span><math><mi>X</mi></math></span>, a Ritt<span><math><msub><mrow></mrow><mrow><mi>E</mi></mrow></msub></math></span> operator admits a quadratic functional calculus if and only if the estimates <span><math><mrow><msub><mrow><mo>‖</mo><","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 5","pages":"Pages 1417-1452"},"PeriodicalIF":0.8,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895930","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 : 2025-06-28DOI: 10.1016/j.indag.2025.05.010
Michael J. Schlosser
Using Bailey’s very-well-poised summation, we show that a specific sequence of well-poised bilateral basic hypergeometric series form a family of orthogonal functions on the unit circle. We further extract a bilateral matrix inverse from Dougall’s summation which we use, in combination with the Pfaff–Saalschütz summation, to derive a summation for a particular bilateral hypergeometric series. We finally provide multivariate extensions of the bilateral matrix inverse and the summation in the setting of hypergeometric series associated to the root system .
{"title":"A family of orthogonal functions on the unit circle and a new multilateral matrix inverse","authors":"Michael J. Schlosser","doi":"10.1016/j.indag.2025.05.010","DOIUrl":"10.1016/j.indag.2025.05.010","url":null,"abstract":"<div><div>Using Bailey’s very-well-poised <span><math><mrow><msub><mrow></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow></math></span> summation, we show that a specific sequence of well-poised bilateral basic hypergeometric <span><math><mrow><msub><mrow></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>ψ</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> series form a family of orthogonal functions on the unit circle. We further extract a bilateral matrix inverse from Dougall’s <span><math><mrow><msub><mrow></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span> summation which we use, in combination with the Pfaff–Saalschütz summation, to derive a summation for a particular bilateral hypergeometric <span><math><mrow><msub><mrow></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> series. We finally provide multivariate extensions of the bilateral matrix inverse and the <span><math><mrow><msub><mrow></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> summation in the setting of hypergeometric series associated to the root system <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1762-1778"},"PeriodicalIF":0.8,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374398","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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