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Let $S_6$ be the symmetric group on 6 letters and let $\alpha \colon S_6 \to S_6$ be an outer automorphism (note that $S_6$ is the only permutation group that has an outer automorphism and that $\mathrm{Out}(S_6) \cong \mathbb{Z}/2\mathbb{Z}$). For any irreducible representation $\rho \colon S_6 \to \mathrm{GL}(V)$ of $S_6$, the composition $\rho \circ \alpha \colon S_6 \to \mathrm{GL}(V)$ is also an irreducible representation.

Question: Is there a reference describing the action of this operation ($\rho \mapsto \rho \circ \alpha$) on the set of all irreducible representations of $S_6$ over $\mathbb{C}$ (that is on the set of partitions of 6)?

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    $\begingroup$ See mathoverflow.net/questions/215298/… $\endgroup$ – Mark Wildon Aug 23 '17 at 22:53
  • $\begingroup$ @MarkWildon: Dear Mark, thanks for your answer, but my question was actually about a reference where this action is described. Don't you know one by chance? $\endgroup$ – Sasha Aug 24 '17 at 7:38
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    $\begingroup$ From the action on conjugacy classes it is very easy to get the action on characters, as I'm sure you know. But despite some searching (a few years ago) I have never seen this written down in a published paper. $\endgroup$ – Mark Wildon Aug 24 '17 at 23:20
  • $\begingroup$ @MarkWildon: Mark, thanks a lot for your help! $\endgroup$ – Sasha Aug 25 '17 at 5:31
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First, notice that this operation preserves the dimension of the representation, this already considerably restricts things: the dimensions of the irreps. are 1,1,5,5,5,5,9,9,10,10,16. The trivial rep. is fixed, thus the sign rep. must also be. The 16-dimensional rep. $V_{(3,2,1)}$ is also fixed.

Next, since $\alpha$ maps conjugacy classes to conjugacy classes, it preserves the multiset of character values of $\rho$. Checking this against a character table shows that the 9-dimensional reps. are also all fixed. This does allow for the possibility that the 5-dimensional rep. $V_{(5,1)}$ is sent to $V_{(2,2,2)}$ and its conjugate $V_{(2,1,1,1,1)}$ is sent to $V_{(3,3)}$. You can see that one of these pairs is swapped if and only if the other is by tensoring with the sign representation. The 10-dimensional irreps. $V_{(4,1,1)}$ and $V_{(3,1,1,1)}$ may also be swapped.

EDIT: After I have written this, Mark Wildon has linked to a question which subsumes this one. His answer there, using an explicit description of $\alpha$'s action on conjugacy classes, is that all of the possible swaps I identified do in fact occur.

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  • $\begingroup$ Dear Christian, thanks for your answer, but my question was actually about a reference where this action is described. Don't you know one by chance? $\endgroup$ – Sasha Aug 24 '17 at 7:39
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    $\begingroup$ @Sasha The automorphism itself and its action on conjugacy classes is described well on Wikipedia. Using arguments like mine or Mark Wildon's on the other question then determines the action on irreps. $\endgroup$ – Christian Gaetz Aug 24 '17 at 12:12
  • $\begingroup$ Thanks for pointing this, there are some references in the Wikipedia article, maybe I will find something there. $\endgroup$ – Sasha Aug 24 '17 at 13:21

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