I am interested in polynomials with icosahedral symmetry.
https://arxiv.org/pdf/1308.0955.pdf
says that
$$
p(v,w)=v^{11}w+11v^6w^6-vw^{11}
$$
has icosahedral symmetry. Each term is homogeneous of degree $12$. Moreover, within each term the power of $v$ and the power of $w$ are congruent mod $ 5 $. That implies $ p $ is invariant under the following generator of the binary icosahedral subgroup of $ SU_2 $
$$
\begin{bmatrix}
e^{\frac{2\pi i}{5}} & 0\\
0 & e^{-\frac{2\pi i}{5}}
\end{bmatrix}=
\begin{bmatrix}
\zeta_5 & 0\\
0 & \overline{\zeta_5}
\end{bmatrix}
$$
In other words, the symmetry
\begin{align*}
v \mapsto \zeta_5 v \\
w \mapsto \overline{\zeta_5} w
\end{align*}
fixes $ p $. Apparently $ p $ is also invariant under
$$
\begin{bmatrix}
-\zeta_5+\overline{\zeta_5} & \zeta_5^2-\overline{\zeta_5}^2\\
\zeta_5^2-\overline{\zeta_5}^2 & \zeta_5-\overline{\zeta_5}
\end{bmatrix}
$$
and these two matrices generate the entire 120 element binary icosahedral subgroup $ 2I \cong SL_2(\mathbb{F}_5) $ of $ SU_2 $.
Are there any lower degree polynomials in two complex variables $ v,w $ with icosahedral symmetry? How about polynomials in three real variables $ x,y,z $ that are invariant with respect to the icosahedral subgroup $ I \cong A_5 \cong PSL_2(\mathbb{F}_5) $ of $ SO_3(\mathbb{R})$?
Best Answer
No, there are no invariants of smaller degree.
Let $G = PSL_2(\mathbb{F}_5)$. Assuming that you are working over a field of characteristic $0$ (in fact, it would be enough to work over a field of characteristic not dividing $\# G$), it is a theorem of Klein, proved in his book on the Icosahedron, that the invariant ring $K[v,w]^G$ is generated by the polynomials \begin{gather*} v^{11} w - 11v^6w^6 - vw^{11}\\ v^{20} + 228 v^{15} w^5 + 494 v^{10} w^{10} - 228 v^5 w^{15} + w^{20} \\ v^{30} - 522 v^{25} w^5 - 10005 v^{20} w^{10} - 10005 v^{10} w^{20} + 522 v^5 w^{25} + w^{30} \end{gather*}
This can be proved by computing the Poincare series using Molien's formula (or by computer algebra e.g., Magma's
FundamentalInvariants).