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

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 $f\le 5$, 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.