The coinvariant algebra of S_2 and S_3 as graded regular representations

Assume the Axiom of Choice, inherited from the published items used. We work over $\mathbb{k}=\mathbb{C}$ and compute the coinvariant algebra of The coinvariant algebra of a finite Weyl group in the two smallest cases and read off its graded decomposition, checking the general theorem The coinvariant algebra is the graded regular representation of W; the example is a verification in small rank and is not used as evidence for the general statement.

The case $W=S_2$. Here $r=1$, the auxiliary polynomial ring is $\mathbb{k}[x]$ with the transposition acting by $x\mapsto -x$, and the basic invariant is $x^2$, so $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$ in the page normalization and $\mathrm{deg}x=1$ for the classical Hilbert series. The Hilbert series is therefore $1+t^2$ in page degrees (equivalently $1+t$ in classical degrees), agreeing with ${\prod }_i(1+t^2+\cdots +t^{2(d_i-1)})$ for the single basic degree $d_1=2$ of Weyl coinvariant hilbert series has order w dimension. The degree-zero piece $C_0=\mathbb{k}$ is the trivial representation, and the one-dimensional top piece $C_2=\mathbb{k}x$ is the sign representation, so $C$ is the regular representation of $S_2$ with graded multiplicities one for the trivial and one for the sign module.

The case $W=S_3$. Use the auxiliary permutation model of The coinvariant algebra of a finite Weyl group: $C=\mathbb{k}[x_1,x_2,x_3]\mathrm{/}(e_1,e_2,e_3),e_1=x_1+x_2+x_3,e_2=x_1{x}_2+x_1{x}_3+x_2{x}_3,e_3=x_1{x}_2{x}_3,$ with $S_3$ permuting the variables. In classical degrees the basic degrees are $1,2,3$, so the Hilbert series is $\frac{(1-t)(1-t^2)(1-t^3)}{(1-t{)}^3}=(1+t)(1+t+t^2)=1+2t+2t^2+t^3,$ and in the page normalization, where each variable has degree $2$, it becomes $(1+t^2)(1+t^2+t^4)=1+2t^2+2t^4+t^6$. The graded pieces have dimensions $1,2,2,1$ in degrees $0,2,4,6$.

Graded pieces as $S_3$-modules. Identify $C$ with the coinvariant algebra of the two-dimensional reflection representation $\mathfrak{h}\subset {\mathbb{k}}^3$. The degree-zero piece $C_0$ is the trivial module. The degree-two piece $C_2$ is spanned by the images of $x_1,x_2,x_3$ modulo $e_1$; these span a two-dimensional space on which $S_3$ acts as the standard reflection representation. Put $a=x_1$, $b=x_2$, $x_3=-a-b$. The relations are $a^2+ab+b^2=0$ and $a^{2}b+a{b}^2=0$, so $b^3=0$ and $C_6$ is spanned by the nonzero class $a{b}^2$ (its nonvanishing also follows from the Hilbert series). The Vandermonde reduces to $-6a{b}^2$, so it spans $C_6$ and carries the sign character. The perfect multiplication pairing $C_2\times C_4\to C_6$ from The coinvariant algebra is a complete intersection with one-dimensional socle identifies $C_4$ with $C_2^{\ast }\otimes \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}$. The standard reflection representation is self-dual: the invariant coordinate dot product restricts nondegenerately to the sum-zero plane. Hence $C_4$ is standard tensor sign. Thus $C_0=\text{trivial},C_2=\text{standard},C_4=\text{standard}\otimes \text{sign},C_6=\text{sign},$ the graded lift of the regular representation, with graded multiplicities $1$ for the trivial module, $t^6$ for the sign module and $t^2+t^4$ for the two-dimensional standard module.

Character check. By The coinvariant algebra is the graded regular representation of W the graded character is $\mathrm{HTr}(C,t)(w)={\prod }_{i=1}^r(1-t^{2d_i})\det (1-t^{2}w\mid {\mathfrak{h}}^{\ast }{)}^{-1}\mathrm{.}$ For a transposition in $S_3$ the eigenvalues on ${\mathfrak{h}}^{\ast }$ are $1,-1$, so $\det (1-t^{2}w)=(1-t^2)(1+t^2)=1-t^4$ and the quotient is $(1-t^4)(1-t^6)\mathrm{/}(1-t^4)=1-t^6$, matching the direct sum $1+0\cdot t^2+0\cdot t^4+(-1)t^6$. For a three-cycle the eigenvalues are the primitive cube roots $\omega ,{\omega }^2$, so $\det (1-t^{2}w)=1+t^2+t^4$ and the quotient is $(1-t^4)(1-t^6)\mathrm{/}(1+t^2+t^4)=(1-t^2)(1-t^4)=1-t^2-t^4+t^6$, matching $1+(-1)t^2+(-1)t^4+t^6$. At $w=1$ the quotient is $1+2t^2+2t^4+t^6$ and its value at $t=1$ is $6=\mathrm{\mid }{S}_3\mathrm{\mid }$. In both cases the graded character at $t=1$ is the character of the regular representation, and the sum of graded multiplicities times dimensions is $1+2+2+1=6=\mathrm{\mid }W\mathrm{\mid }$, respectively $1+1=2=\mathrm{\mid }{S}_2\mathrm{\mid }$, as the general theorem asserts.