What does the regular representation of the coinvariant ring of a unitary reflection group look like?

Question

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.

Answer 1

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.

Answer 2

I think if the question is "give an explicit constuction" vs "give a nice construction", then an answer could go along these lines:

1. Pick a basis $\{ f_i\}$ for the coinvariant ring.
2. Construct the standard isotypic idempotents of the group. These are the central elements of the group algebra defined as $$e_{\chi}:=\dfrac{\chi(1)}{|W|}\sum_{w\in W}\chi(w^{-1})w,$$ for any irreducible character $\chi$ of $W$.
3. Use the idempotents and your basis $\{f_i\}$ from (1) to construct bases for the different isotypic components $U_{\chi}$ in the coinvariant algebra. That is, for each $\chi$ as above, find a $\mathbb{C}$-linear basis of the span $U_{\chi}:=\langle e_{\chi}\cdot f_i\rangle$.
4. Break each isotypic component to $\chi(1)$-many $W$-modules. That is easy to do; just pick an element $f\in U_{\chi}$ and find a $\mathbb{C}$-linear basis of $\{ w\cdot f\,|\,w\in W\}$. Then proceed with the remaining elements of $U_{\chi}$ etc. Many times the $\chi$-exponents are all different which means you can identify the different $W$-modules just by degree considerations.

5. Now you are left with the following problem: You have two explicitly given $W$-modules (i.e. with given $\mathbb{C}$-bases and an explicit $W$-action on them) that are isomorphic and you need to construct the isomorphism. This should come down to a linear system that would do an ambient change of basis so that the two matrix representations for a given set of generators would agree.

For the part (5) above you can also do (in most cases) something analogous to the Okounkov-Vershik method which can give a natural basis for each irreducible module. You would need the Jucys-Murphy elements in the symmetric group and any multiplicity-free tower of subgroups for the other groups. In all real cases apart from E8 (I think) you can get such a tower only by using reflection subgroups.