Show that the sum over a group of the matrix elements of any irreducible representation other than the identity representation is equal to zero.

This is the exercise in a group theory textbook, the question is the exactly as the book presents. Thanks for help.


For all $h$ in $G$, $$\sum_{g{\rm\ in\ }G}\rho(g)=\sum_{g{\rm\ in\ }G}\rho(gh)=\sum_{g{\rm\ in\ }G}\rho(g)\rho(h)=\left(\sum_{g{\rm\ in\ }G}\rho(g)\right)\rho(h)$$ so it reduces to showing that $I-\rho(h)$ is invertible for some $h$ in $G$.

  • $\begingroup$ Thanks. But if for all h in G, $I-\rho(h)$ is not invertible, how to deduce a contradiction? $\endgroup$ – yangcs11 May 20 '18 at 11:48
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    $\begingroup$ Perhaps easier to note that this calculation shows that the given sum spans a subrepresentation on which the group acts trivially. $\endgroup$ – Tobias Kildetoft May 20 '18 at 13:39
  • $\begingroup$ @Tobias, Oh, I see! Thank you very much. $\endgroup$ – yangcs11 May 21 '18 at 0:22

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