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Good Basic Invariants and Frobenius Structures

Ikuo Satake

Abstract

In this paper, we define a set of good basic invariants for a finite complex reflection group under certain conditions. We show that a set of good basic invariants for a finite real reflection group gives a set of the flat invariants obtained by Saito and the Taylor coefficients of these good basic invariants give the structure constants of the multiplication of the Frobenius structure obtained by Dubrovin.

Good Basic Invariants and Frobenius Structures

Abstract

In this paper, we define a set of good basic invariants for a finite complex reflection group under certain conditions. We show that a set of good basic invariants for a finite real reflection group gives a set of the flat invariants obtained by Saito and the Taylor coefficients of these good basic invariants give the structure constants of the multiplication of the Frobenius structure obtained by Dubrovin.

Paper Structure

This paper contains 23 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Aim and results of the paper
  3. Acknowledgements
  4. Graded ${\mathbb C}$-algebra structure on $\mathbb{C}[V]$
  5. Graded ${\mathbb C}$-algebra structure on $\mathbb{C}[V]^G$
  6. Admissible triplet
  7. Graded ${\mathbb C}$-algebra structure on $\mathbb{C}[V]$
  8. Graded ${\mathbb C}$-algebra Isomorphism $\psi$
  9. The morphism $\varphi[g,\zeta,q]$
  10. The morphism $\psi[g,\zeta,q]$
  11. Good basic invariants
  12. Examples
  13. Independence on the choice of the admissible triplet
  14. Taylor coefficients of the good basic invariants
  15. The cases of finite real reflection groups
  16. ...and 8 more sections

Key Result

Proposition 3.2

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 3.1
  • Proposition 3.2
  • Proof
  • Definition 3.3
  • Proposition 3.4
  • Proof
  • Definition 3.5
  • ...and 37 more