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A reduction theorem for good basic invariants of finite complex reflection groups

Yukiko Konishi, Satoshi Minabe

TL;DR

The paper introduces and develops a reduction framework for Satake's good basic invariants of finite complex reflection groups through δ-reflection subquotients $G_{δ}$, showing that when the largest degree $d_1$ is regular and $δ|d_1$, a good basic invariants set for $G$ induces one for $G_{δ}$. It proves a Reduction Theorem linking admissible triplets and graded coordinates between $G$ and $G_{δ}$, and shows that the potential vector field of a duality group $G$ reduces to that of $G_{δ}$ as well. The authors illustrate the construction with explicit sequences for several families, including $G(m,m,n+1)$, $G(m,1,n)$, and the exceptional $E_6$ via $G_{35}$, demonstrating concrete maps of invariants and potentials across successive subquotients, such as $E_6\to F_4\to G_{25}\to G_8\to G_5$ and $G_{31}\to G_9$, $G_{10}$. These results provide a systematic pathway to understand flat coordinates, Saito structures, and multiplication constants on orbit spaces under reduction, enriching the theory of duality structures in complex reflection groups.

Abstract

This is a sequel to our previous article arXiv:2307.07897. We describe a certain reduction process of Satake's good basic invariants. We show that if the largest degree $d_1$ of a finite complex reflection group $G$ is regular and if $δ$ is a divisor of $d_1$, a set of good basic invariants of $G$ induces that of the reflection subquotient $G_δ$. We also show that the potential vector field of a duality group $G$, which gives the multiplication constants of the natural Saito structure on the orbit space, induces that of $G_δ$. Several examples of this reduction process are also presented.

A reduction theorem for good basic invariants of finite complex reflection groups

TL;DR

The paper introduces and develops a reduction framework for Satake's good basic invariants of finite complex reflection groups through δ-reflection subquotients , showing that when the largest degree is regular and , a good basic invariants set for induces one for . It proves a Reduction Theorem linking admissible triplets and graded coordinates between and , and shows that the potential vector field of a duality group reduces to that of as well. The authors illustrate the construction with explicit sequences for several families, including , , and the exceptional via , demonstrating concrete maps of invariants and potentials across successive subquotients, such as and , . These results provide a systematic pathway to understand flat coordinates, Saito structures, and multiplication constants on orbit spaces under reduction, enriching the theory of duality structures in complex reflection groups.

Abstract

This is a sequel to our previous article arXiv:2307.07897. We describe a certain reduction process of Satake's good basic invariants. We show that if the largest degree of a finite complex reflection group is regular and if is a divisor of , a set of good basic invariants of induces that of the reflection subquotient . We also show that the potential vector field of a duality group , which gives the multiplication constants of the natural Saito structure on the orbit space, induces that of . Several examples of this reduction process are also presented.
Paper Structure (33 sections, 6 theorems, 139 equations, 2 figures)

This paper contains 33 sections, 6 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regular elements, reflection subquotients, admissible triplets and good basic invariants
  3. Notations
  4. Regular elements
  5. Reflection subquotients in the regular cases
  6. Admissible triplet and good basic invariants
  7. Reduction Theorem
  8. Admissible triplet and reflection subquotients
  9. Good basic invariants of $G_{\delta}$
  10. Potential vector field of $G_{\delta}$
  11. Reflection subquotients of $G(m, m, n+1)$ and $G(m,1,n)$
  12. $G(m,m,n+1)$ for $m\geq 2$, $n\geq 1$
  13. $G(m,1,n)$ for $m\geq 2$
  14. The reflection subquotient of $G(m,m,n+1)$ by $\delta=m$
  15. Reflection subquotients of $G(m,1,n)$
  16. ...and 18 more sections

Key Result

Proposition 2.3

Let $G$ be a finite complex reflection group. Let $\delta$ be a regular number for $G$, i.e. a positive integer satisfying $\mathfrak{a}(\delta)=\mathfrak{b}(\delta)$. Let $h\in G$ be a $\xi$-regular element with respect to a $\delta$-th primitive root of unity $\xi$ and let $E=V(h,\xi)$. Then the f

Figures (2)

  • Figure 1: The sequence of reflection subquotients of $G_{35}=E_6$. All those six groups are duality groups with $d_1=12$. The numbers written under each group are the degrees of that group.
  • Figure 2: The sequence of reflection subquotients of $G_{31}$. Note that $G_{31}$ is not a duality group while the other three groups are duality groups with $d_1=24$.

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 3.1
  • ...and 14 more