Indagationes Mathematicae-New Series最新文献

{"title":"Minimal cubature rules and Koornwinder polynomials","authors":"Yuan Xu","doi":"10.1016/j.indag.2025.06.002","DOIUrl":"10.1016/j.indag.2025.06.002","url":null,"abstract":"<div><div>In his classical paper (Koornwinder, 1974), Koornwinder studied a family of orthogonal polynomials of two variables, derived from symmetric polynomials. This family possesses a rare property that orthogonal polynomials of degree <span><math><mi>n</mi></math></span> have <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span> real common zeros, which leads to important examples in the theory of minimal cubature rules. This paper aims to give an account of the minimal cubature rules of two variables and examples originating from Koornwinder polynomials, and we will also provide further examples.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1863-1878"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374402","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":"Bounds for asymptotic characters of simple Lie groups","authors":"Pavel Etingof ,&nbsp;Eric Rains","doi":"10.1016/j.indag.2025.02.003","DOIUrl":"10.1016/j.indag.2025.02.003","url":null,"abstract":"<div><div>An important function attached to a complex simple Lie group <span><math><mi>G</mi></math></span> is its asymptotic character <span><math><mrow><mi>X</mi><mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> (where <span><math><mrow><mi>λ</mi><mo>,</mo><mi>x</mi></mrow></math></span> are real (co)weights of <span><math><mi>G</mi></math></span>) – the Fourier transform in <span><math><mi>x</mi></math></span> of its Duistermaat–Heckman function <span><math><mrow><mi>D</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> (continuous limit of weight multiplicities). It is shown in Garibaldi et al. that the best <span><math><mi>λ</mi></math></span>-independent upper bound <span><math><mrow><mo>−</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><msub><mrow><mo>inf</mo></mrow><mrow><mi>x</mi></mrow></msub><mi>Re</mi><mi>X</mi><mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> for fixed <span><math><mi>λ</mi></math></span> is strictly negative. We quantify this result by providing a lower bound for <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> in terms of <span><math><mrow><mo>dim</mo><mi>G</mi></mrow></math></span>. We also provide upper and lower bounds for <span><math><mrow><mi>D</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>λ</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math></span> when <span><math><mrow><mrow><mo>|</mo><mi>λ</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span>. This allows us to show that <span><math><mrow><mrow><mo>|</mo><mi>X</mi><mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≤</mo><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>λ</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> for some constant <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> depending only on <span><math><mi>G</mi></math></span>, which implies the conjecture in Remark 17.16 of Garibaldi et al. We also show that <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>≤</mo><msup><mrow><mrow><mo>(</mo><mfrac><mrow><mn>4</mn></mrow><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span>. Finally, in the appendix we prove Conjecture 1 in Coquereaux and Zuber (2018) about Mittag-Leffler type sums for <span><math><mi>G</mi></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1572-1591"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374410","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 partial-sum deformationfor a family of orthogonal polynomials","authors":"Erik Koelink ,&nbsp;Pablo Román ,&nbsp;Wadim Zudilin","doi":"10.1016/j.indag.2025.04.007","DOIUrl":"10.1016/j.indag.2025.04.007","url":null,"abstract":"<div><div>There are several questions one may ask about polynomials <span><math><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>m</mi></mrow></msubsup><msup><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msup><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> attached to a family of orthogonal polynomials <span><math><msub><mrow><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of <span><math><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>;</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> in the case of varying real parameter <span><math><mi>t</mi></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1745-1761"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374397","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":"Boundary transfer matrices arising from quantum symmetric pairs","authors":"Andrea Appel ,&nbsp;Bart Vlaar","doi":"10.1016/j.indag.2025.05.008","DOIUrl":"10.1016/j.indag.2025.05.008","url":null,"abstract":"<div><div>We introduce a universal framework for boundary transfer matrices, inspired by Sklyanin’s two-row transfer matrix approach for quantum integrable systems with boundary conditions. The main examples arise from quantum symmetric pairs of finite and affine type. As a special case we recover a construction by Kolb in finite type. We review recent work on universal solutions to the reflection equation and highlight several open problems in this field.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1830-1862"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374401","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 versatility of the Drinfeld double of a finite group","authors":"Giovanna Carnovale ,&nbsp;Nicola Ciccoli ,&nbsp;Elena Collacciani","doi":"10.1016/j.indag.2025.03.003","DOIUrl":"10.1016/j.indag.2025.03.003","url":null,"abstract":"<div><div>The Drinfeld double of a finite group appears in many different areas of mathematics and physics<span><span>. We review different instances in which the Drinfeld double of a finite group and its representations play a role, focusing on some of Tom Koornwinder’s research interests: harmonic analysis, Lie algebras, </span>quantum groups, non-commutative geometry, and Verlinde formula for fusion rules.</span></div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1600-1627"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374686","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":"Exercises on the Kepler ellipses through a fixed point in space, after Otto Laporte","authors":"G.J. Heckman","doi":"10.1016/j.indag.2025.02.004","DOIUrl":"10.1016/j.indag.2025.02.004","url":null,"abstract":"<div><div>This article has a twofold purpose. On the one hand I would like to draw attention to some nice exercises on the Kepler laws, due to Otto Laporte from 1970. Our discussion here has a more geometric flavor than the original analytic approach of Laporte.</div><div>On the other hand it serves as an addendum to a paper of mine from 1998 on the quantum integrability of the Kovalevsky top. Later I learned that this integrability result had been obtained already long before by Laporte in 1933.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1592-1599"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374346","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":"Boundedness of the Cherednik kernel and its limit transition from type BC to type A","authors":"Dominik Brennecken","doi":"10.1016/j.indag.2025.04.004","DOIUrl":"10.1016/j.indag.2025.04.004","url":null,"abstract":"<div><div>We introduce a Cherednik kernel and a hypergeometric function for integral root systems and prove their relation to spherical functions associated with Riemannian symmetric spaces of reductive Lie groups. Furthermore, we characterize the spectral parameters for which the Cherednik kernel is a bounded function. In the case of a crystallographic root system, this characterization was proven by Narayanan, Pasquale and Pusti for the hypergeometric function. This result generalizes the Helgason–Johnson theorem from 1969, which characterizes the bounded spherical functions of a Riemannian symmetric space. The characterization for the Cherednik kernel is based on recurrence relations for the associated Cherednik operators under the dual affine Weyl group going back to Sahi. These recurrence relations are also used to prove a limit transition between the Cherednik kernel of type <span><math><mi>A</mi></math></span> and of type <span><math><mi>B</mi></math></span>, which generalizes an already known result for the associated hypergeometric functions by Rösler, Koornwinder, and Voit.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1717-1744"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374395","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 tale of two q-deformations: Connecting dual polar graphs and weighted hypercubes","authors":"Pierre-Antoine Bernard ,&nbsp;Étienne Poliquin ,&nbsp;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":"2025-11-01","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}

{"title":"Automorphisms of the DAHA of type C1ˇC1 and non-symmetric Askey–Wilson functions","authors":"Tom Koornwinder ,&nbsp;Marta Mazzocco","doi":"10.1016/j.indag.2025.05.005","DOIUrl":"10.1016/j.indag.2025.05.005","url":null,"abstract":"<div><div>In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type <span><math><mrow><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>ˇ</mo></mrow></mover><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> which have a relatively simple action on the generators and on the parameters, notably a symmetry <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> which sends the Askey–Wilson (AW) parameters <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></math></span> to <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>q</mi><msup><mrow><mi>d</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>q</mi><msup><mrow><mi>c</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></math></span>. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric AW polynomials and functions. Interestingly <span><math><msub><mrow><mi>t</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> maps AW polynomials to functions. We take the rank one case of Stokman’s Cherednik kernel for <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> as the definition of the non-symmetric Askey–Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1795-1829"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374400","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 S3-symmetric q-Onsager algebra and its Lusztig automorphisms","authors":"Paul Terwilliger","doi":"10.1016/j.indag.2025.02.001","DOIUrl":"10.1016/j.indag.2025.02.001","url":null,"abstract":"<div><div>The <span><math><mi>q</mi></math></span>-Onsager algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is defined by two generators and two relations, called the <span><math><mi>q</mi></math></span>-Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, now called the Lusztig automorphisms. Recently, we introduced a generalization of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> called the <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-symmetric <span><math><mi>q</mi></math></span>-Onsager algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. The algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> has six distinguished generators, said to be standard. The standard <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy the <span><math><mi>q</mi></math></span>-Dolan/Grady relations. In the present paper we do the following: (i) for each standard <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-generator we construct an automorphism of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> called a Lusztig automorphism; (ii) we describe how the six Lusztig automorphisms of <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are related to each other; (iii) we describe what happens if a finite-dimensional irreducible <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-module is twisted by a Lusztig automorphism; (iv) we give a detailed example involving an irreducible <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-module with dimension 5.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 6","pages":"Pages 1533-1554"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145374407","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}