Return to Answer

Actually you may find Professor Stanley's paper on invariant theory useful (in particular Proposition 4.9). So here's one idea: for a fixed finite reflection group $W$, and a fixed linear character $\chi\colon W\rightarrow{\mathbb C}^*$, let $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$ be its relative invariant ring, and let $f_{\chi}\in R^W_\chi$ be a non-zero generator (for reflection groups, $R^W_\chi$ is a cyclic $R^W$ module). Let $\chi_1,\ldots,\chi_m$ be the list of distinct linear characters of the group $W$.

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

Proof: Assume that $W$ is abelian and consider the action of $W$ on $R_W={\mathbb C}[V]^{coW}$. Since $W$ is abelian there is a simultaneous eigenbasis $\left\{v_1,\ldots,v_m\right\}\subset R_W$ (so $m=|W|$). Since $R_W$ is the regular representation for $W$, each basis element $v_i$ corresponds to a distinct linear character $\chi_i\colon W\rightarrow{\mathbb C}^*$ (and they all show up). One can show that each $v_i$ lifts to a relative invariant $f_i\in R^W_{\chi_i}$ (e.g. take any lift, and average over the kernel of $\chi_i$, I think). To see that the (equivalence classes of the) polynomials $\left\{\left.\sum_{i=1}^m\chi_i(w)f_i\right|w\in W\right\}$ are linearly independent in $R_W$, use linear independence of distinct linear characters.

For non-abelian groups you'll need something more complicated I think...

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.

Actually you may find Professor Stanley's paper on invariant theory useful (in particular Proposition 4.9). So here's one idea: for a fixed finite reflection group $W$, and a fixed linear character $\chi\colon W\rightarrow{\mathbb C}^*$, let $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$ be its relative invariant ring, and let $f_{\chi}\in R^W_\chi$ be a non-zero generator (for reflection groups, $R^W_\chi$ is a cyclic $R^W$ module). Let $\chi_1,\ldots,\chi_m$ be the list of distinct linear characters of the group $W$.

Conjecture: The elements $\left\{\sum_{i=1}^m\chi_i(w)\cdot f_{\chi_i}\left|w\in W\right.\right\}$ form a basis on which $W$ acts by the regular representation.

It seems to work for your $S_2$ example anyway...

Edit: This conjecture is not true if the the group has more elements than linear characters.

Actually you may find Professor Stanley's paper on invariant theory useful (in particular Proposition 4.9). So here's one idea: for a fixed finite reflection group $W$, and a fixed linear character $\chi\colon W\rightarrow{\mathbb C}^*$, let $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$ be its relative invariant ring, and let $f_{\chi}\in R^W_\chi$ be a non-zero generator (for reflection groups, $R^W_\chi$ is a cyclic $R^W$ module). Let $\chi_1,\ldots,\chi_m$ be the list of distinct linear characters of the group $W$.

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

Proof: Assume that $W$ is abelian and consider the action of $W$ on $R_W={\mathbb C}[V]^{coW}$. Since $W$ is abelian there is a simultaneous eigenbasis $\left\{v_1,\ldots,v_m\right\}\subset R_W$ (so $m=|W|$). Since $R_W$ is the regular representation for $W$, each basis element $v_i$ corresponds to a distinct linear character $\chi_i\colon W\rightarrow{\mathbb C}^*$ (and they all show up). One can show that each $v_i$ lifts to a relative invariant $f_i\in R^W_{\chi_i}$ (e.g. take any lift, and average over the kernel of $\chi_i$, I think). To see that the (equivalence classes of the) polynomials $\left\{\left.\sum_{i=1}^m\chi_i(w)f_i\right|w\in W\right\}$ are linearly independent in $R_W$, use linear independence of distinct linear characters.

For non-abelian groups you'll need something more complicated I think...

Actually you may find Professor Stanley's paper on invariant theory useful (in particular Proposition 4.9). So here's one idea: for a fixed finite reflection group $W$, and for a fixed character $\chi\colon W\rightarrow{\mathbb C}^*$, let $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$ be its relative invariant ring, and let $f_{\chi}\in R^W_\chi$ be a non-zero generator (for reflection groups, $R^W_\chi$ is a cyclic $R^W$ module). Let $\chi_1,\ldots,\chi_m$ be the list of irreducible characters of the group $W$.

Conjecture: The elements $\left\{\sum_{i=1}^m\chi_i(w)\cdot f_{\chi_i}\left|w\in W\right.\right\}$ form a basis on which $W$ acts by the regular representation.

It seems to work for your $S_2$ example anyway...

Actually you may find Professor Stanley's paper on invariant theory useful (in particular Proposition 4.9). So here's one idea: for a fixed finite reflection group $W$, and a fixed linear character $\chi\colon W\rightarrow{\mathbb C}^*$, let $R^W_\chi=\left\{g\in R\left|w\cdot g=\chi(w)\cdot g, \ \forall w\in W\right.\right\}$ be its relative invariant ring, and let $f_{\chi}\in R^W_\chi$ be a non-zero generator (for reflection groups, $R^W_\chi$ is a cyclic $R^W$ module). Let $\chi_1,\ldots,\chi_m$ be the list of distinct linear characters of the group $W$.

Conjecture: The elements $\left\{\sum_{i=1}^m\chi_i(w)\cdot f_{\chi_i}\left|w\in W\right.\right\}$ form a basis on which $W$ acts by the regular representation.

It seems to work for your $S_2$ example anyway...

Edit: This conjecture is not true if the the group has more elements than linear characters.