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

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.