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

TL;DR

This paper studies the complex reflection group related to the octahedral group, computing all irreducible representations, character tables, vector-valued invariants, and explicit dimension formulas for invariant rings.

Contribution

It provides a complete analysis of the irreducible representations and invariants of a specific complex reflection group associated with the octahedral group, including explicit formulas.

Findings

All irreducible representations of the group are determined.

Character table of the group is computed.

Explicit dimension formulas for invariant rings are derived.

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.