Satake's good basic invariants for finite complex reflection groups

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2025-07-18 DOI:10.1016/j.geomphys.2025.105597

Yukiko Konishi , Satoshi Minabe

引用次数: 0

Abstract

In [17] Satake introduced the notions of admissible triplets and good basic invariants for finite complex reflection groups. For irreducible finite Coxeter groups, he showed the existence and the uniqueness of good basic invariants. Moreover he showed that good basic invariants are flat in the sense of K. Saito's flat structure. He also obtained a formula for the multiplication of the Frobenius structure. In this article, we generalize his results to finite complex reflection groups. We first study the existence and the uniqueness of good basic invariants. Then for duality groups, we show that good basic invariants are flat in the sense of the natural Saito structure constructed in [7]. We also give a formula for the potential vector fields of the multiplication in terms of the good basic invariants. Moreover, in the case of irreducible finite Coxeter groups, we derive a formula for the potential functions of the associated Frobenius manifolds.

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有限复反射群的Satake良好的基本不变量

1996年，Satake引入了有限复反射群的可容许三重和良好基本不变量的概念。对于不可约有限Coxeter群，他证明了良好基本不变量的存在唯一性。此外，他还证明了好的基本不变量在齐藤的平坦结构意义上是平坦的。他还得到了Frobenius结构的乘法公式。在本文中，我们将他的结果推广到有限复反射群。我们首先研究了好的基本不变量的存在唯一性。然后，对于对偶群，我们证明了良好的基本不变量在[7]中构造的自然Saito结构意义上是平坦的。我们也给出了一个关于乘法的势向量场的公式，用好的基本不变量表示。此外，在不可约有限Coxeter群的情况下，我们导出了相关Frobenius流形的势函数公式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity