Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ for which $\operatorname{dim}(\operatorname{ker}(r-1)) = n-1$.

It is well-known that a finite subgroup of $GL(V)$ is a unitary reflection group if and only if $\mathbb{C}[V]^W \cong \mathbb{C}[f_1,\ldots,f_n]$ is again a polynomial algebra generated by homogeneous polynomials $f_1,\ldots,f_n$.

In this case, it is also well known that the coinvariant algebra $\mathbb{C}[V]^{\operatorname{coW}} = \mathbb{C}[V] \big/ \langle f_1,\ldots,f_n\rangle$ carries the regular representation as a $W$-module. This is, every $d$-dimensional irreducible representation of $W$ is found $d$ times inside $\mathbb{C}[V]^{\operatorname{coW}}$.

All this can be found e.g. in the Lehrer-Taylor book "Unitary reflection groups" in Chapter 3.

In Proposition 3.23, the result about the regular representation is obtained by showing that $\mathbb{C}[V]^{\operatorname{coW}}$ has dimension $|W|$ and the character vanishes for elements $\neq 1 \in W$.

My question now is

Is it known how to actually construct the $W$-equivariant isomorphisms between an irreducible representation $\Lambda$ of $W$ and its $\operatorname{dim}(\Lambda)$ many copies inside $\mathbb{C}[V]^{\operatorname{coW}}$ ?

A pointer to any treatment (being it only the symmetric group, real reflection groups, or more general situations) would be much appreciated!

Btw: It is also known how to compute the homogeneous degrees in which the copies of $\Lambda$ sit based on the character values of $\chi_\Lambda$ using the description of the fake degree as found e.g. in Lemma 4.21 in the reference. So this is not part of the question, only how to get the explicit isomorphisms.


I believe this question is answered (at least for the case of the symmetric group) by the "higher Specht basis" of Ariki, Terasoma, and Yamada; see the recent preprint Gillespie and Rhoades - Higher Specht bases for generalizations of the coinvariant ring for a discussion of this problem in a broader context.

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