Consider the action of a finite group $G$ by conjugation. What is the character of the corresponding permutation representation $\mathbb C G$? Prove that the sum of elements in any row of the character table of $G$ is a non-negative integer.

For the first part, there's a lemma that says $\chi_{\mathbb CX}(g) = | \{x \in X \ | \ gx = x \}| $. In this case $X = G$ and we have that $\{ x \in X \ | \ gx = x \} = \{ x \in X \ | \ g x g^{-1} = x \} = C_G(g)$, so that $\chi_{\mathbb CX}(g) = |C_G(g)|$.

I'm having trouble with the second part. There's another lemma that says $\langle 1, \chi_{\mathbb C X} \rangle$ is the number of orbits of $G$ on $X$, which in this case is the number of conjugacy classes (i.e. the number of irreducible representations of $G$). So we know that the character $\chi_{\mathbb C G}$ contains $n$ copies of the trivial character in its decomposition, where $n$ is the number of conjugacy classes / representations of the group. I'm not sure if this is helpful, or where to go from here.

Any hints would be greatly appreciated. Thanks

  • $\begingroup$ Take another character $\chi'$. Then $\sum _{i=1}^r \chi'(x_i) = \sum_{i=1}^r \frac{\overline{\chi(x_i)}\chi'(x_i)}{|C_g(x_i)|} $ (where the $x_i$ are conjugacy class representatives). Does this look familiar? $\endgroup$ – Daniel Freedman Mar 20 '12 at 23:38

Hint: compute the inner product of an arbitrary character with $\chi_{\mathbb{C}[G]}$. Break up the sum into sums over conjugacy classes.


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