Given a group representation, how can I definitely know whether it is irreducible or not? In principle I should check for non-trivial invariant subspaces, or find, if any, block-triangular similar matrix, but that sounds computationally difficult.
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If you are working over $\mathbb{C}$, for example, there is a very nice computational way to check. Namely, suppose that you have a representation $\rho$. You can decompose this into a direct sum of irreps $\displaystyle \bigoplus \rho_\alpha$. If $\chi_\alpha$ is the character corresponding to $\rho_\alpha$ and $\chi_\rho$ the character corresponding to $\rho$ you can see that $$\langle\chi_\rho,\chi_\rho\rangle=\sum_\alpha n_\alpha^2$$ And, since $\rho$ will be irreducible if and only if there is one irrep in this decomposition, which corresponds to one $n_\alpha$ being $1$ and the rest zero, which corresponds to $\displaystyle \sum_\alpha n_\alpha^2=1$ you may conclude that $\rho$ is irreducible if and only if $\langle\chi_\rho,\chi_\rho\rangle=1$ |
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