Let \[ A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. \]
Show that \(A\) generates a cyclic group \(G\) of order \(4\). Show that if \(f\in\mathbb{C}[x,y]^G\), then \(f\) can have no monomials of odd degree.
Use Molien’s Theorem to show that the Hilbert series of \(G\) is \[ \phi_G(z)=\frac{1+z^4}{(1-z^2)(1-z^4)}. \]
Show that \(\mathbb{C}[x,y]^G\) is Cohen–Macaulay by explicitly finding the generators and separators for \(\mathbb{C}[x,y]^G\).
Solution
Goal: Analyze the ring of invariants of the order-4 rotation.
1. Generate the action and part (a):
1.1. \(G=\langle A\rangle\) with\[A=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix},\qquad A^4=I,\]so \(|G|=4\) and\[A(x,y)=(-y,x).\]
1.2. For a monomial \(x^iy^j\),\[A\cdot x^iy^j=(-y)^i x^j=x^j y^i(-1)^i.\]
1.3. The element \(A^2=-I\) sends \(x^iy^j\mapsto(-1)^{i+j}x^iy^j\).
1.4. Invariance under \(A^2\) therefore forces \(i+j\) even, so no invariant monomial has odd total degree. Hence part (a).
2. Compute the Hilbert series (part (b)):
2.1. Molien gives\[\phi_G(z)=\frac1{4}\sum_{g\in G}\frac{1}{\det(I-zg)}.\]
2.2. The four determinants are:\[\det(I-zI)= (1-z)^2,\]\[\det(I-zA)=1+z^2,\]\[\det(I-zA^2)=(1+z)^2,\]\[\det(I-zA^3)=1+z^2.\]
2.3. Therefore\[\phi_G(z)=\frac14\left(\frac{1}{(1-z)^2}+\frac{2}{1+z^2}+\frac{1}{(1+z)^2}\right) =\frac{1+z^4}{(1-z^2)(1-z^4)}.\]
3. Part (c), Cohen–Macaulay structure:
3.1. Use coordinates\[u=x+iy,\qquad v=x-iy.\]Then\[A\cdot u = i u,\qquad A\cdot v = -i v.\]
3.2. A monomial \(u^av^b\) is invariant iff \(a-b\equiv0\pmod4\).
3.3. Put\[p=u^4=(x+iy)^4,\qquad q=v^4=(x-iy)^4,\qquad r=uv=x^2+y^2.\]Then every invariant is a polynomial in \(p,q,r\) with relation\[pq=r^4,\]so\[\mathbb C[x,y]^G\cong \mathbb C[p,q,r]/(pq-r^4).\]
3.4. Set \(s=p+q\) and \(t=p\). The elements \(r=uv\) and \(s=u^4+v^4\) are algebraically independent: their Jacobian determinant with respect to \(u,v\) is\[\det\begin{pmatrix}v&u\\4u^3&4v^3\end{pmatrix}=4(v^4-u^4),\]which is not the zero polynomial. Since the ground field has characteristic \(0\), any algebraic relation between \(r\) and \(s\) would force the Jacobian to have rank \(<2\) identically. Thus \(S=\mathbb C[r,s]\) is a polynomial ring in two variables.
3.5. In the presentation \(\mathbb C[p,q,r]/(pq-r^4)\), write \(q=s-t\) and \(p=t\). Then\[t^2-st+r^4=0,\]so\[\mathbb C[x,y]^G\cong S[t]/(t^2-st+r^4).\]Because the defining polynomial is monic of degree \(2\) in \(t\), Euclidean division in \(S[t]\) gives a unique representative\[a(r,s)+t\,b(r,s),\qquad a,b\in S.\]Hence\[\mathbb C[x,y]^G=S\oplus tS\]as an \(S\)-module.
3.6. Therefore \(\mathbb C[x,y]^G\) is a free module over the polynomial subring \(S=\mathbb C[r,s]\), hence Cohen–Macaulay, with homogeneous system of parameters \(r\) (degree \(2\)), \(s\) (degree \(4\)), and secondary invariants (the requested separators) \(\{1,t\}\).
4. Conclusion: All requested properties hold, and the stated Hilbert series is proved. Q.E.D.