A2 coinvariant algebra and basic invariants

Each transposition fixes a line in $V$ and has eigenvalue $-1$ on the vector that subtracts its two exchanged coordinates. These transpositions generate $S_3$. The action is faithful: a permutation fixing all coordinate differences fixes their labels and hence is identity. Eliminate $z=-x-y$ to identify its polynomial algebra with $\mathbb{C}[x,y]$ and obtain $e_2=-(x^2+xy+y^2)$ and $e_3=-xy(x+y)$.

We verify generation directly. Any invariant polynomial on the plane has a polynomial lift to $\mathbb{C}[x,y,z]$; average its six permuted lifts. By F2 its restriction is unchanged and the resulting lift is symmetric. A symmetric homogeneous polynomial in three variables is a polynomial in $e_1=x+y+z,e_2,e_3$: use lexicographic order $x>y>z$ on its finitely many monomials of fixed total degree. A leading exponent triple $(a,b,c)$ satisfies $a\ge b\ge c$, because swapping an out-of-order pair would produce a larger monomial with the same coefficient. The polynomial $e_1^{a-b}{e}_2^{b-c}{e}_3^c$ has leading monomial $x^a{y}^b{z}^c$ with coefficient one. Subtract its required multiple and repeat; lexicographic order strictly decreases within a finite set. Sum over the finitely many homogeneous parts. Restricting $e_1=0$ proves invariant generation by $e_2,e_3$.

These generators are algebraically independent without an appeal to a degree table. Their Jacobian determinant in $(x,y)$ is $(2x+y)(x+2y)(x-y)$, a nonzero polynomial. If a nonzero relation of least positive total degree $H(e_2,e_3)=0$ existed, differentiating it twice, once in each coordinate, and multiplying by the adjugate Jacobian matrix would give that determinant times each evaluated partial derivative of $H$ is zero. The polynomial ring is a domain, so both evaluated derivatives vanish. In characteristic zero some partial derivative of a nonconstant $H$ is nonzero and has lower degree, a contradiction; a nonzero constant cannot be a relation. Thus the stated invariant polynomial algebra has basic degrees two and three.

By 2.1 and 3.1, the positive invariant ideal in F2 generates $(e_2,e_3)$ in $\mathbb{C}[x,y]$. Put $a=y^2+xy+x^2$ and $b=xy(x+y)$. The identity $xa-b=x^3$ gives $(a,b)=(a,x^3)$. First quotient by $x^3$: with $A=\mathbb{C}[x]\mathrm{/}(x^3)$, the remaining quotient is $A[y]\mathrm{/}(y^2+xy+x^2)$. Division by this monic polynomial in $y$ has unique remainder $u+vy$ with $u,v\in A$. Existence follows by canceling the highest power of $y$; uniqueness follows because a nonzero multiple of a monic degree-two polynomial has degree at least two, even over $A$. The unique representatives of elements of $A$ have degrees at most two in $x$. Hence $1,x,x^2,y,xy,x^{2}y$ are independent and spanning, proving the asserted basis and Hilbert series by their degrees $0,1,2,1,2,3$.

The quotient dimension is therefore $6=\mathrm{\mid }{S}_3\mathrm{\mid }$, and the degree product is $2\cdot 3=6$. The degree-zero class is nonzero, the top class $x^{2}y$ is nonzero by unique remainder, and all degrees above three vanish. This is an explicit calculation on the nonzero two-dimensional plane; there is no limiting parameter or choice assumption.