= Solution
A <group representation> on a <vector space> $V$ is a <group homomorphism> $R:G\to\operatorname{GL}(V)$, so $R(gh)=R(g)R(h)$ and $R(e)=I$. A nonzero <group representation> is an <irreducible representation> if its only <invariant subspaces> are $0$ and $V$. Here the <representations> and <characters> are finite-dimensional over the <complex numbers>.
The <character of a representation> is $\chi_R(g)=\operatorname{tr}R(g)$. Three useful properties are:
* It is a <class function>: $\chi_R(hgh^{-1})=\chi_R(g)$, by the cyclic property of the <trace>.
* It is additive on a <direct sum> and multiplicative on a <tensor product of group representations>: $\chi_{R\oplus S}=\chi_R+\chi_S$ and $\chi_{R\otimes S}=\chi_R\chi_S$.
* The <irreducible characters> obey <character orthogonality>:
$$
\langle\chi_r,\chi_s\rangle=\frac1{|G|}\sum_{g\in G}\chi_r(g)^*\chi_s(g)=\delta_{rs}.
$$
In particular, $\chi_R(e)=\dim V$ and $\chi_R(g^{-1})=\chi_R(g)^*$, the latter following from <unitarization of a finite-group representation>. The <character inner product> $\langle\chi_r,\chi_R\rangle$ counts the multiplicity of an <irreducible representation> $r$ in $R$.
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