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Vector-Valued Invariants Associated with All Irreducible Representations for a Finite Group

A. K. M. Selim Reza, Manabu Oura, Masashi Kosuda

Abstract

We investigate the complex reflection group $\mathfrak{G}$ associated with the octahedral group, identified as the ninth entry in the Shephard-Todd classification. We determine all irreducible representations of $\mathfrak{G}$ and compute the character table. Moreover, for each representation, we compute the module of vector-valued invariants and relate it to the fundamental invariants of the octahedral group. Additionally, we derive explicit dimension formulas for the corresponding rings of invariants.

Vector-Valued Invariants Associated with All Irreducible Representations for a Finite Group

Abstract

We investigate the complex reflection group associated with the octahedral group, identified as the ninth entry in the Shephard-Todd classification. We determine all irreducible representations of and compute the character table. Moreover, for each representation, we compute the module of vector-valued invariants and relate it to the fundamental invariants of the octahedral group. Additionally, we derive explicit dimension formulas for the corresponding rings of invariants.
Paper Structure (4 sections, 4 theorems, 55 equations, 3 tables)

This paper contains 4 sections, 4 theorems, 55 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Irreducible Representations of $\mathfrak{G}$
  3. Vector-valued invariants
  4. Results

Key Result

Theorem 1

The eight one-dimensional representations are indexed by pairs $(a,b)$ with $a\in\{0,1,2,3\}$ and $b\in\{0,1\}$, and module $M(\rho_{a,b})$ is a free $\mathfrak{R}$-module of rank $1$ generated by $\Gamma^{a}\Delta^{b}$: In particular, for each $\rho_i$ with $i=1,2,3,\dots,8$, the generators are with the corresponding dimension formulas of the form

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4