Computation of Vector-Valued Invariants for a Finite Complex Reflection Group

TL;DR

This paper computes vector-valued invariants for a specific complex reflection group and explores the structure of its invariant rings, advancing understanding in invariant theory for complex reflection groups.

Contribution

It provides explicit computations of invariants for a particular complex reflection group and analyzes the structure of its invariant rings, which is a novel contribution.

Findings

Explicit invariants computed for various representations.

Structural insights into the invariant rings obtained.

Enhanced understanding of complex reflection group invariants.

Abstract

We consider the complex reflection group \( \mathcal{G} \), identified as No. 8 in the Shephard-Todd classification. In this paper, we present computations of the vector-valued invariants associated with various representations of \( \mathcal{G} \). Additionally, we investigate the structure of the corresponding invariant rings.