On the Structure of Multilinear Invariants of a Finite Unitary Reflection Group

TL;DR

This paper investigates the structure of multilinear invariants in finite unitary reflection groups, explicitly determining bases and their relationships for degrees up to five, revealing connections to Catalan numbers.

Contribution

It introduces a subspace of typical invariants, calculates dimensions using Catalan numbers, and explicitly constructs bases for degrees up to five.

Findings

Dimension of subspace W_f equals Catalan number

Explicit bases for invariants up to degree 5

Relationship between bases of V_f and W_f established

Abstract

We study the space of multilinear invariants \( V_f \) of degree \( f \) for a specified finite unitary reflection group. A subspace \( W_f \) of typical invariants is also introduced. We note that the dimension of \( W_f \) is given by Catalan number. We explore both spaces for \( f \leq 5 \), noting that their dimensions differ based on the value of \( f \). We explicitly determine the bases for both spaces, and then we establish the relationship between the vectors of the two bases.