What are the polynomials in two real variables that are invariant under the action of \(D_4\), the symmetry group of a square, by rotations and reflections on the plane that the two variables form?
Solution
Let \(D_4\) act on \(\RR[x,y]\) through the symmetries of the square with vertices \((\pm1,\pm1)\) in the \((x,y)\)-plane. Then \[ \RR[x,y]^{D_4}=\RR[x^2+y^2,\,x^2y^2]. \]
1. Every \(D_4\)-invariant polynomial lies in \(\RR[x^2,y^2]\).
\(D_4\) contains the reflections \((x,y)\mapsto(-x,y)\) and \((x,y)\mapsto(x,-y)\). If \(f=\sum_{i,j}a_{ij}x^iy^j\) is fixed by the first, then \(a_{ij}=0\) for every odd \(i\); invariance under the second similarly gives \(a_{ij}=0\) for every odd \(j\). Hence \(f(x,y)=F(x^2,y^2)\) for some \(F\in\RR[u,v]\).
2. A polynomial \(F(x^2,y^2)\) is \(D_4\)-invariant if and only if \(F(u,v)=F(v,u)\).
\(D_4\) contains the reflection \((x,y)\mapsto(y,x)\) in a diagonal, so invariance gives \(F(x^2,y^2)=F(y^2,x^2)\). The substitution \(u\mapsto x^2\), \(v\mapsto y^2\) is an injective homomorphism \(\RR[u,v]\to\RR[x,y]\), so this is equivalent to \(F(u,v)=F(v,u)\). Conversely, the two sign changes and the interchange of coordinates generate the group of \(2\times2\) signed permutation matrices, which has order \(8\) and consists of symmetries of the square, so it is all of \(D_4\). A polynomial \(F(x^2,y^2)\) with \(F\) symmetric is fixed by each of these generators, hence by \(D_4\).
3. \(\RR[u,v]^{S_2}=\RR[u+v,\,uv]\).
This is the fundamental theorem of symmetric polynomials in two variables.
4. \(\RR[x,y]^{D_4}=\RR[x^2+y^2,\,x^2y^2]\).
By 1 and 2, an invariant is \(F(x^2,y^2)\) with \(F\) symmetric; by 3, \(F\) is a polynomial in \(u+v\) and \(uv\), and substituting \(u=x^2\), \(v=y^2\) expresses the invariant as a polynomial in \(x^2+y^2\) and \(x^2y^2\). Conversely, \(x^2+y^2\) and \(x^2y^2\) are invariant by 2, so every polynomial in them is invariant.
5. The pair \(x^2+y^2\), \(x^4+y^4\) generates the same ring.
\[ x^4+y^4=(x^2+y^2)^2-2x^2y^2, \qquad x^2y^2=\frac{(x^2+y^2)^2-(x^4+y^4)}{2}, \] and \(2\) is invertible in \(\RR\).