Indagationes Mathematicae-New Series最新文献

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Minimal cubature rules and Koornwinder polynomials 最小培养规则和Koornwinder多项式

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-06-07 DOI: 10.1016/j.indag.2025.06.002

Yuan Xu

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

n

have

n (n + 1) / 2

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.

在他的经典论文（Koornwinder, 1974）中，Koornwinder研究了由对称多项式导出的二元正交多项式族。该族具有n次正交多项式有n(n+1)/2个实数公零的罕见性质，这是最小培养规则理论中的重要例子。本文给出了两个变量的最小培养规则，并给出了源于Koornwinder多项式的例子，我们还将提供进一步的例子。

引用次数: 0

Bounds for asymptotic characters of simple Lie groups 单李群的渐近特征的界

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-02-25 DOI: 10.1016/j.indag.2025.02.003

Pavel Etingof , Eric Rains

An important function attached to a complex simple Lie group

G

is its asymptotic character

X (λ, x)

(where

λ, x

are real (co)weights of

G

) – the Fourier transform in

x

of its Duistermaat–Heckman function

D H_{λ} (p)

(continuous limit of weight multiplicities). It is shown in Garibaldi et al. that the best

λ

-independent upper bound

- c (G)

for

{inf}_{x} Re X (λ, x)

for fixed

λ

is strictly negative. We quantify this result by providing a lower bound for

c (G)

in terms of

dim G

. We also provide upper and lower bounds for

D H_{λ} (0)

when

| λ | = 1

. This allows us to show that

| X (λ, x) | \leq C (G) {| λ |}^{- 1} {| x |}^{- 1}

for some constant

C (G)

depending only on

G

, which implies the conjecture in Remark 17.16 of Garibaldi et al. We also show that

c (S L_{n}) \leq {(\frac{4}{π^{2}})}^{n - 2}

. Finally, in the appendix we prove Conjecture 1 in Coquereaux and Zuber (2018) about Mittag-Leffler type sums for

G

附在复单李群G上的一个重要函数是它的渐近特征X(λ， X)（其中λ， X是G的实（co）权）-它的Duistermaat-Heckman函数DHλ(p)（权复数的连续极限）在X中的傅里叶变换。Garibaldi等人证明，对于固定λ的infxReX(λ,x)，最佳λ无关上界- c(G)是严格负的。我们通过用dimG给出c(G)的下界来量化这个结果。当|λ|=1时，我们还给出了DHλ(0)的上界和下界。这允许我们证明|X(λ， X)|≤C(G)|λ|−1| X |−1对于某常数C(G)只依赖于G，这暗示了Garibaldi等人在备注17.16中的猜想。我们还证明了c(SLn)≤(4π2)n−2。最后，在附录中，我们证明了Coquereaux和Zuber（2018）关于G的Mittag-Leffler型和的猜想1。

引用次数: 0

A partial-sum deformationfor a family of orthogonal polynomials 正交多项式族的部分和变形

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-05-09 DOI: 10.1016/j.indag.2025.04.007

Erik Koelink , Pablo Román , Wadim Zudilin

There are several questions one may ask about polynomials

q_{m} (x) = q_{m} (x; t) = \sum_{n = 0}^{m} t^{n} p_{n} (x)

attached to a family of orthogonal polynomials

{p_{n} (x)}_{n \geq 0}

. 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

q_{m} (x; t)

in the case of varying real parameter

t

对于正交多项式族{pn(x)}n≥0的多项式qm(x)=qm(x;t)=∑n=0mtnpn(x)，有几个问题。在本笔记中，我们提请注意这种部分和变形的自然性和相关的美丽结构。特别地，我们研究了在实参数t变化的情况下qm（x;t）的零点的位置和分布。

引用次数: 0

Boundary transfer matrices arising from quantum symmetric pairs 由量子对称对引起的边界转移矩阵

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-06-06 DOI: 10.1016/j.indag.2025.05.008

Andrea Appel , Bart Vlaar

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.

受Sklyanin的具有边界条件的量子可积系统的两行转移矩阵方法的启发，我们引入了边界转移矩阵的通用框架。主要的例子来自有限型和仿射型的量子对称对。作为特例，我们恢复了有限型的Kolb构造。我们回顾了最近关于反射方程的全称解的工作，并强调了该领域的几个开放问题。

引用次数: 0

The versatility of the Drinfeld double of a finite group 有限群的德林菲尔德双元的多功能性

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-03-26 DOI: 10.1016/j.indag.2025.03.003

Giovanna Carnovale , Nicola Ciccoli , Elena Collacciani

The Drinfeld double of a finite group appears in many different areas of mathematics and physics. 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, quantum groups, non-commutative geometry, and Verlinde formula for fusion rules.

有限群的德林菲尔德双元出现在数学和物理的许多不同领域。我们回顾了有限群的德林菲尔德双元及其表示所起作用的不同实例，重点介绍了Tom Koornwinder的一些研究兴趣：调和分析、李代数、量子群、非交换几何和融合规则的Verlinde公式。

引用次数: 0

Exercises on the Kepler ellipses through a fixed point in space, after Otto Laporte 以奥托·拉波特（Otto Laporte）的名字命名，在空间中的一个固定点上练习开普勒椭圆

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-03-07 DOI: 10.1016/j.indag.2025.02.004

G.J. Heckman

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.

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.

这篇文章有双重目的。一方面，我想让大家注意一些关于开普勒定律的很好的练习，这是1970年Otto Laporte提出的。我们这里的讨论比拉波特原来的解析方法更有几何色彩。另一方面，它是我1998年关于Kovalevsky顶的量子可积性的论文的补充。后来我才知道，这个可积性结果早在1933年拉波特就已经得到了。

引用次数: 0

Boundedness of the Cherednik kernel and its limit transition from type BC to type A Cherednik核的有界性及其从BC型到A型的极限跃迁

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-04-28 DOI: 10.1016/j.indag.2025.04.004

Dominik Brennecken

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

A

and of type

B

, which generalizes an already known result for the associated hypergeometric functions by Rösler, Koornwinder, and Voit.

引入了积分根的Cherednik核和超几何函数，并证明了它们与约化李群的黎曼对称空间中的球函数的关系。进一步，我们刻画了Cherednik核为有界函数的谱参数。对于晶体根系统，Narayanan、Pasquale和Pusti用超几何函数证明了这一特性。这个结果推广了1969年的Helgason-Johnson定理，该定理描述了黎曼对称空间的有界球函数。Cherednik核的表征是基于对偶仿射Weyl群下相关Cherednik算子的递归关系，该递归关系可以追溯到Sahi。这些递归关系还用于证明a型Cherednik核与B型Cherednik核之间的极限跃迁，推广了Rösler、Koornwinder和Voit对相关超几何函数的已知结果。

引用次数: 0

A tale of two q-deformations: Connecting dual polar graphs and weighted hypercubes 两个q-变形：连接对偶极图和加权超立方体

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-07-08 DOI: 10.1016/j.indag.2025.05.011

Pierre-Antoine Bernard , Étienne Poliquin , Luc Vinet

Two

q

-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

U_{q} (su (2))

and of the dual

q

-Krawtchouk polynomials are elaborated upon. This paper is dedicated to Tom Koornwinder.

介绍了超立方图的两个q-类似物，并通过图商证明了它们之间的关系。讨论了子空间格图、Uq（su(2)）的扭曲元和对偶q-Krawtchouk多项式的作用。本文献给Tom Koornwinder。

引用次数: 0

Automorphisms of the DAHA of type C1ˇC1 and non-symmetric Askey–Wilson functions C1 / C1型DAHA的自同构与非对称Askey-Wilson函数

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-05-29 DOI: 10.1016/j.indag.2025.05.005

Tom Koornwinder , Marta Mazzocco

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type

\overset{ˇ}{C_{1}} C_{1}

which have a relatively simple action on the generators and on the parameters, notably a symmetry

t_{4}

which sends the Askey–Wilson (AW) parameters

(a, b, c, d)

(a, b, q d^{- 1}, q c^{- 1})

. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric AW polynomials and functions. Interestingly

t_{4}

maps AW polynomials to functions. We take the rank one case of Stokman’s Cherednik kernel for

B C_{n}

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.

在本文中，我们考虑了类型为C1 + C1的双仿射Hecke代数（DAHA）的自同构，它们对产生子和参数有相对简单的作用，特别是一个对称t4，它将Askey-Wilson （AW）参数（a,b,c,d）发送到（a,b,qd−1,qc−1）。我们研究了这些对称性如何作用于基本表示以及对称和非对称的AW多项式和函数。有趣的是，t4将AW多项式映射为函数。我们取BCn的Stokman的Cherednik核的秩一情况作为非对称Askey-Wilson函数的定义。由此我们得到一个表达式，它是一个对称项和一个反对称项的和。

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引用次数: 0

The S3-symmetric q-Onsager algebra and its Lusztig automorphisms s3对称q-Onsager代数及其Lusztig自同构

IF 0.8 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2025-11-01 Epub Date: 2025-02-11 DOI: 10.1016/j.indag.2025.02.001

Paul Terwilliger

The

q

-Onsager algebra

O_{q}

is defined by two generators and two relations, called the

q

-Dolan/Grady relations. In 2019, Baseilhac and Kolb introduced two automorphisms of

O_{q}

, now called the Lusztig automorphisms. Recently, we introduced a generalization of

O_{q}

called the

S_{3}

-symmetric

q

-Onsager algebra

O_{q}

. The algebra

O_{q}

has six distinguished generators, said to be standard. The standard

O_{q}

-generators can be identified with the vertices of a regular hexagon, such that nonadjacent generators commute and adjacent generators satisfy the

q

-Dolan/Grady relations. In the present paper we do the following: (i) for each standard

O_{q}

-generator we construct an automorphism of

O_{q}

called a Lusztig automorphism; (ii) we describe how the six Lusztig automorphisms of

O_{q}

are related to each other; (iii) we describe what happens if a finite-dimensional irreducible

O_{q}

-module is twisted by a Lusztig automorphism; (iv) we give a detailed example involving an irreducible

O_{q}

-module with dimension 5.

q-Onsager代数Oq由两个生成器和两个关系定义，称为q-Dolan/Grady关系。2019年，Baseilhac和Kolb引入了Oq的两个自同构，现在称为Lusztig自同构。最近，我们介绍了Oq的一种推广，称为s3对称q-Onsager代数Oq。代数Oq有六个不同的生成器，据说是标准的。标准的oq生成器可以用正六边形的顶点来标识，使得非相邻生成器可交换，相邻生成器满足q-Dolan/Grady关系。在本文中，我们做了以下工作：(i)对于每个标准的Oq生成器，我们构造了一个Oq的自同构，称为Lusztig自同构；（ii）描述了Oq的六个Lusztig自同构之间的相互关系；（iii）描述了有限维不可约oq模被Lusztig自同构扭曲时的情形；（iv）我们给出了一个涉及维数为5的不可约oq模的详细例子。

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引用次数: 0

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