Let \(G\) be the subgroup of \(S_4\) generated by the permutations \((12)\) and \((34)\), and let \(V\) be the simple representation of \(S_4\) labeled by the Young diagram \([2,1,1]\).
Calculate the Molien series of \(k[x_1, x_2, x_3]^G\).
Let \(\tilde{G} \cong \mathbb{Z}/2 \times \mathbb{Z}/2\) with generators \(g_1\) and \(g_2\), and let \(W\) be the three-dimensional \(\tilde{G}\)-module with basis \(w_1, w_2, w_3\) such that the action of \(\tilde{G}\) is given by \[g_1 w_i = \begin{cases} -w_i & \text{if } i = 1, 2 \\ w_3 & \text{if } i = 3 \end{cases} \qquad g_2 w_i = \begin{cases} w_1 & \text{if } i = 1 \\ -w_i & \text{if } i = 2, 3 \end{cases}\] Write down a Hironaka decomposition for \(k[y_1, y_2, y_3]^{\tilde{G}}\).
Find presentations of the rings \(k[y_1, y_2, y_3]^{\tilde{G}}\) and \(k[x_1, x_2, x_3]^G\) via generators and relations.
Solution
Assume the coefficient field has characteristic \(0\).
Part (a).
1. The restriction of the \(S_4\)-module \(S^{(2,1,1)}\) to\[ G=\langle(12),(34)\rangle\cong C_2\times C_2 \]has character values\[ \chi(e)=3, \qquad \chi(g)=-1\quad(g\ne e). \]
The representation \(S^{(2,1,1)}\) is the sign twist of the standard representation \(S^{(3,1)}\). For the standard representation of \(S_4\), \[ \chi_{(3,1)}(\sigma)=\#\operatorname{Fix}(\sigma)-1. \] A transposition has two fixed points, so its standard character value is \(1\); twisting by sign gives \(-1\). The element \((12)(34)\) has no fixed points and sign \(+1\), so its value is also \[ 0-1=-1. \] The identity has value \(3\).
2. Every nonidentity element of \(G\) acts on \(V\) with eigenvalues\[ 1,-1,-1. \]
Each nonidentity element has order \(2\), so its eigenvalues are \(\pm1\). Their sum is the character value \(-1\) from 1, in dimension \(3\). Hence exactly one eigenvalue is \(1\) and two are \(-1\).
3. The Molien series is\[ M_G(t) =\frac14\left( \frac1{(1-t)^3} +\frac3{(1-t)(1+t)^2} \right) =\frac{1+t^3}{(1-t^2)^3}. \]
Molien’s formula gives \[ M_G(t)=\frac1{|G|}\sum_{g\in G}\frac1{\det(I-tg)}. \] The identity contributes \((1-t)^{-3}\). By 2, each of the three nonidentity elements contributes \[ \frac1{(1-t)(1+t)^2}. \] Combining the terms and simplifying gives \[ \frac14\left( \frac1{(1-t)^3}+ \frac3{(1-t)(1+t)^2} \right) =\frac{1+t^3}{(1-t^2)^3}. \]
Part (b).
4. A monomial \(y_1^{a_1}y_2^{a_2}y_3^{a_3}\) is \(\widetilde G\)-invariant if and only if\[ a_1\equiv a_2\equiv a_3\pmod2. \]
The generator \(g_1\) changes the sign of \(y_1\) and \(y_2\), so invariance under \(g_1\) is equivalent to \[ a_1+a_2\equiv0\pmod2. \] Likewise, \(g_2\) changes the signs of \(y_2,y_3\), so invariance under \(g_2\) is equivalent to \[ a_2+a_3\equiv0\pmod2. \] These two congruences say precisely that \(a_1,a_2,a_3\) have the same parity.
5. A Hironaka decomposition is\[ k[y_1,y_2,y_3]^{\widetilde G} =k[y_1^2,y_2^2,y_3^2] \oplus y_1y_2y_3\,k[y_1^2,y_2^2,y_3^2]. \]
By 4, every invariant monomial has either all three exponents even or all three odd. In the first case it lies in \(k[y_1^2,y_2^2,y_3^2]\). In the second it is \(y_1y_2y_3\) times a monomial in the three squares. The two summands have disjoint monomial bases, so the sum is direct.
Part (c).
6. The first invariant ring has presentation\[ k[y_1,y_2,y_3]^{\widetilde G} \cong k[a,b,c,d]/(d^2-abc), \]where\[ a=y_1^2, \quad b=y_2^2, \quad c=y_3^2, \quad d=y_1y_2y_3. \]
5. Hence the induced homomorphism has trivial kernel beyond \((d^2-abc)\) and is surjective.
7. The restricted \(G\)-module \(V\) is isomorphic to the \(\widetilde G\)-module \(W\) in part (b), after relabeling the three nontrivial characters.
Since \(G\) is abelian of exponent \(2\), its complex representations split into one-dimensional characters. Let its four characters be \(1,\chi_1,\chi_2,\chi_3\). The character in 1 equals \[ \chi_1+\chi_2+\chi_3: \] at the identity this sum is \(3\), while at every nonidentity element exactly one of the three nontrivial characters is \(1\) and the other two are \(-1\), so the sum is \(-1\). Thus \[ V|_G\cong\chi_1\oplus\chi_2\oplus\chi_3. \] The module \(W\) has exactly the same three nontrivial characters by the sign actions in part (b). Hence the two representations are isomorphic.
8. Consequently, after choosing an eigenbasis \(z_1,z_2,z_3\) for \(V|_G\) corresponding to the three nontrivial characters,\[ k[x_1,x_2,x_3]^G \cong k[z_1,z_2,z_3]^G \cong k[A,B,C,D]/(D^2-ABC), \]with\[ A=z_1^2, \quad B=z_2^2, \quad C=z_3^2, \quad D=z_1z_2z_3. \]
A linear change of coordinates induced by the \(G\)-module isomorphism of 7 identifies the polynomial representations and hence their invariant rings. The calculation of <1>4–<1>6 then applies verbatim in the eigenbasis \(z_1,z_2,z_3\). The Hilbert series of this hypersurface is \[ \frac{1-t^6}{(1-t^2)^3(1-t^3)} =\frac{1+t^3}{(1-t^2)^3}, \] in agreement with the Molien series in 3.