Basis of coinvariant algebra on which reflection group acts as regular representation

Question

This question is almost a duplicate of a question of Christian Stump, except that Christian seems to ask about an isomorphism to irreducible representations rather than the regular representation: What does the regular representation of the coinvariant ring of a unitary reflection group look like?

If $W\subseteq GL(V)$ is a finite reflection group, then the Chevalley-Shephard-Todd theorem says that the invariant algebra $\mathbb{C}[V]^W$ is isomorphic to a polynomial algebra $\mathbb{C}[V]^W\simeq \mathbb{C}[e_1,e_2,\ldots,e_n]$ and that the coinvariant algebra $\mathbb{C}[V]^{\mathrm{co}W} := \mathbb{C}[V]/\mathbb{C}[V]^W_+$ (where $\mathbb{C}[V]^W_+$ is the set of invariant polynomials of positive degree) is isomorphic as a $W$-module to the left regular representation.

Question: can we write down some explicit basis of $\mathbb{C}[V]^{\mathrm{co}W}$ on which $W$ acts as the regular representation?

The only proofs I know of the fact that $\mathbb{C}[V]^{\mathrm{co}W}$ carries the regular representation use character computations which have a "non-constructive" flavor.

Really I am most interested just in the case of the symmetric group $W=S_n$.

Let me give a quick example of what this looks like for $W=S_2$. Then $\mathbb{C}[V] = \mathbb{C}[x_1,x_2]$ and we get $\mathbb{C}[V]^W = \mathbb{C}[x_1+x_2,x_1x_2]$ (these are the ``elementary symmetric polynomials''). (Maybe strictly speaking because $S_2$ acts on $\mathbb{R}^2/(1,1)$ I should write $\mathbb{C}[V] = \mathbb{C}[x_1,x_2]/\langle x_1+x_2 \rangle$ and $\mathbb{C}[V]^W =\mathbb{C}[x_1x_2]$ but I don't think this technicality matters.) At any rate we have that the coinvariant ring is $\mathbb{C}[V]^{\mathrm{co}W}=\mathbb{C}[x_1,x_2]/\langle x_1+x_2,x_1x_2 \rangle$. There are standard bases of the coinvariant ring for the symmetric group, like the staircase monomials or the Schubert polynomials. In this case both of those bases would be $\{x_1,1\}$ (note that those bases are homogeneous). But the symmetric group $S_2$ does not act on that basis as in the regular representation. Instead I would want a basis like $\{x_1+1,-x_1+1\}$.

EDIT: Christian Gaetz asked me what taking $W=C_n=\langle c \rangle$ the cyclic group of order $n$ looks like. Here $c$ acts on $\mathbb{C}$ by $c\cdot x = \xi x$ with $\xi$ a primitive $n$th root of unity. Thus $\mathbb{C}[V]^{\mathrm{co}W} = \mathbb{C}[x]/\langle x^n \rangle$. A good choice of basis here is $\{c^m\cdot f\colon m=0,1,\ldots,n-1\}$ where $f=1+x+x^2+\cdots+x^{n-1}$. Indeed, in this case you can check that all the $c^m\cdot f$ are linearly independent by writing them in the standard basis $\{1,x,\ldots,x^{n-1}\}$ and then evaluating a Vandermonde determinant where you plug in $x_i=\xi^i$.

Answer 1

Any basis $B$ on which $W$ acts as the regular representation has the form $\{w\cdot f\,\mid\, w\in W\}$ for some $f$ in the coinvariant ring. Hence it suffices to take $f$ generic. It would be interesting to find some "nice" nongeneric $f$.

Answer 2

Let $W$ be a finite reflection group. Set $R={\mathbb C}[V]$, $R^W\subset R$ the invariant subring, and $R_W={\mathbb C}[V]^{coW}$ the coinvariant ring. For a linear character $\chi\colon W\rightarrow {\mathbb C}^*$, define the relative invariant ring $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$. For reflection groups, this is always a cyclic module over $R^W$ (e.g. see Professor Stanley's paper). If $\chi_1,\ldots,\chi_m$ are the distinct linear characters of $W$, let $f_1,\ldots,f_m$ be any generators of the respective relative invariant rings. For $f\in R$, let $\bar{f}\in R_W$ denote its equivalence class.

Claim: If $W$ is abelian, then the elements $\left\{\sum_{i=1}^m\chi_i(w)\cdot \bar{f}_i\left|w\in W\right.\right\}$ form a basis for $R_W$ on which $W$ acts by the regular representation.

Proof: Note that the equivalence classes $\left\{\bar{f}_1,\ldots,\bar{f}_m\right\}$ are linearly independent because each is a simultaneous eigenvector for $W$ corresponding to a distinct linear character. Since $W$ is abelian this must be a basis for $R_W$. To see that the elements $\left\{\left.\sum_{i=1}^m\chi_i(w)\bar{f}_i\right|w\in W\right\}$ are linearly independent in $R_W$, use linear independence of distinct linear characters.

For non-abelian groups the elements $\left\{\bar{f}_1,\ldots,\bar{f}_m\right\}$ won't be a basis (since $m<|W|$), which will force the set $\left\{\sum_{i=1}^m\chi_i(w)\bar{f}_i\left|w\in W\right.\right\}$ to be linearly dependent.