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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})$?

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1 Answer 1

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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).

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  • $\begingroup$ Thanks so much, this is exactly what I wanted! Is there any similar result for the case of $ G=PSL_2(\mathbb{F}_5) $ and the invariant ring $ \mathbb{R}[x,y,z]^G $? Also two minor questions: is there a reason you wrote the icosahedral group $ PSL_2(\mathbb{F}_5) $ in your question when the action seems to be associated to $ SU_2 $ and the binary icosahedral group $ SL_2(\mathbb{F}_5) $? Also the reference I looked at had a positive sign in front of the $ 11v^6w^6 $ while your answer has a negative sign. Is that difference significant? $\endgroup$ Dec 29, 2021 at 21:08
  • $\begingroup$ The difference in the sign comes from me accidentally swapping the roles of $v$ and $w$ (then taking $-p$) - I should have followed your convention, sorry. Writing $PSL_2(\mathbb{F}_5)$ was a mistake, it came from the fact I was thinking geometrically (about the action on $\mathbb{P}^1$) when I wrote this answer, in this context $-I$ acts trivially. $\endgroup$ Dec 29, 2021 at 23:18
  • $\begingroup$ Certainally there will be a similar result for the action on $\mathbb{R}[x,y,z]$, since Hilbert-Noether showed that the invariants are a finitely generated $\mathbb{R}$-algebra. One can work it out once you have an explicit representation of your group - the book of Benson "Polynomial Invariants of Finite Groups" is a good reference. $\endgroup$ Dec 29, 2021 at 23:21
  • $\begingroup$ My representation is just the standard representation of $ SO_3(\mathbb{R}) $ acting in $ \mathbb{R}^3 $ (really the restriction of that action to the icosahedral subgroup). Thanks for the reference I'll look into it! $\endgroup$ Dec 30, 2021 at 15:08
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    $\begingroup$ If you work with the larger group of all symmetries of the icosahedron, including the orientation reversing ones, then the ring of invariants is a polynomial ring. (The ring of invariants for any reflection group is a polynomial ring.) The degrees of the generators for this reflection group are $2$, $6$ and $10$. The additional generator in degree $15$ should be the product of the $15$ hyperplanes which contain a pair of antipodal edges of the icosahedron. $\endgroup$ Mar 11, 2022 at 14:00

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