The alternating group $A\_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations should be known in literature but I am not able to find them. Again the matrix entries are also irrational numbers, so I can't compute the generators for the invariant ring in any of the computer algebra systems. Any reference in this direction is highly appreciated. 



Three dimensional representations of Alternating group
Let $G\_0$ be the image of $A\_5$ under one of the $3$-dimensional 
representations, and $G = \pm G\_0$. Then $G$ is the group of
symmetries of the icosahedron, which is a Euclidean reflection group
(type $H\_3$, Shephard-Todd #23). Thus $G$ has a polynomial invariant group,
and in this case the generator degrees are $2, 6, 10$. For invariants
$\phi\_2, \phi\_6, \phi\_{10}$ we can take the Euclidean norm $x \mapsto (x,x)$, 
the product of six linear forms $x \mapsto (v,x)$ where $\pm v$ ranges over
$6$ pairs of vertices of the icosahedron, and the product of ten
linear forms $x \mapsto (v^\*,x)$ where $\pm v^\*$ ranges over
$10$ pairs of vertices of the dual dodecahedron. The invariant ring
of $G\_0 \cong A\_5$ can then be recovered as 
${\bf C}[\phi\_2, \phi\_6, \phi\_{10}, \phi\_{15}]$ 
where $\phi\_{15}$ is the Jacobian determinant of $\phi\_2, \phi\_6, \phi\_{10}$
(and the product of linear forms $(e,x)$ with $\pm e$ ranging over
pairs of edge centers of either the icosahedron or the dodecahedron);
these generators satisfy one relation of the form
$\phi\_{15}^2 = P(\phi\_2, \phi\_6, \phi\_{10})$.


