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Are homomorphisms from binary polyhedral groups to (simple and simply connected) compact Lie groups classified?

For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie algebras", section 8.6.

From the binary icosahedral to E8, there is a work by Frey.

Are there some results for binary dihedral groups, say?

Any binary polyhedral group comes with a standard embedding into SU(2). You can then compose it with a homomorphism from SU(2) to the compact Lie group, which is characterized by a nilpotent orbit by Jacobson-Morozov, which are of course classified. So this construction gives a nice subset. But how general is it?

Any information would be appreciated.

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    $\begingroup$ "For cyclic groups, the result is well known (see e.g. Kac's "Infinite dimensional Lie algebras", section 8.6)." Not exactly... $\endgroup$
    – Mikhail Borovoi
    Jan 5, 2017 at 16:43
  • $\begingroup$ Victor Kac describes automorphisms of given finite order $r$ of a simple complex Lie algebra $\mathfrak g$ up to conjugation in the automorphism group of $\mathfrak g$, and his method gives you a classification of elements of order $r$ up to conjugation in the adjoint group $G=\mathrm{Inn}(\mathfrak g)$ of "inner" automorphisms of $\mathfrak g$. For a classification of elements of order $r$ up to conjugation in a simply connected simple group see this preprint, Section 3. $\endgroup$
    – Mikhail Borovoi
    Jan 5, 2017 at 17:10
  • $\begingroup$ Homomorphisms to U$(n)$ = unitary representations ($n$ not fixed). Homomorphisms to SU$(n)$ = unitary reps with det 1. This is worse-behaved in general, e.g. decomposition into irreps doesn't preserve being det 1. So, in a sense, sticking to simply connected targets is a bit artificial. $\endgroup$
    – YCor
    Nov 2, 2023 at 16:48

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Yes, each unipotent orbit determines a homomorphism from ${\rm SL}_2$ to $G=G({\mathbb C})$, which we can obviously compose with the inclusion of any finite subgroup $\Gamma\subset{\rm SL}_2$. It's easy to see from Kac's classification that there are homomorphisms from cyclic groups to $G$ which don't arise this way. At the opposite extreme, I looked (somewhat briefly) at Frey's classification of homomorphisms $I\rightarrow G(E_8)$ where $I$ is the binary icosahedral group. It seems that all such homomorphisms arise via restriction of a homomorphism from ${\rm SL}_2$ to $G$. (Frey mentions an "inconsistency" however which may cast some doubt on this.) But different (non-conjugate) homomorphisms ${\rm SL}_2\rightarrow G$ can have the same restriction to $I$: for example, according to Frey's results there is a unique class of homomorphisms $I\rightarrow G$ with trivial connected centralizer in $G$. But there are seven distinguished orbits for which the corresponding homomorphism $I\rightarrow G$ has this property. So these seven homomorphisms ${\rm SL}_2\rightarrow G$ must all have the same restriction to $I$ (which in fact defines a homomorphism from ${\mathcal A}_5$ to $G$, since these orbits are even).

I don't know of any work on homomorphisms from binary dihedral groups (in general) to $G$, but would also be interested if such a classification has appeared.

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