Tensor product of representations

Using formula (6) and the tables from item 1 § 2, item 2 § 5, item 4 §, verify that the decompositions

hold for the tensor square of the two-dimensional representation $\Phi^{(3)}$ of the symmetric group $S_{3}$, and

for the tensor square of the two-dimensional representation $\Phi^{(5)}$ of the quaternion group $Q_{8}$.

Representations of the direct product of groups. Let there be two groups $G, H$ with linear representations $(\Phi , V),(\Psi , W)$. Then, setting

where $g \cdot h$ is an element of the direct product $G \times H$ of the groups $G, H$, we make $G \times H$ act on the tensor product $V \otimes \mathbb {C} W$; as usual,

Verify that the map so defined

is a representation of the group $G \times H$ with character $\chi_{\Phi \otimes \Psi }=\chi_{\Phi } \chi_{\Psi }$. Prove the following statement. Let $\Phi^{(1)}, \ldots , \Phi^{(r)}$ (respectively $\Psi^{(1)}, \ldots , \Psi^{(s)}$) be all the irreducible representations of the group $G$ (respectively $H$). Then the representations $\Phi^{(i)} \otimes \Psi^{(j)}$ of the group $G \times H$ are irreducible, and all the irreducible representations of the group $G \times H$ are exhausted by the representations

The forms $x y, x^{n}+y^{n}$ are invariants of the two-dimensional linear dihedral group

(see Exercise 9 from § 5). Show that any other (polynomial) invariant of the group $\left(D_{n}, \Phi \right)$ has the form of a polynomial in $x y, x^{n}+y^{n}$.

Verify that the quaternion group, considered in its two-dimensional irreducible representation, has no quadratic or cubic invariants. What can be said about the forms $x^{2} y^{2}, x^{4}+y^{4}$?