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Let $ G $ be a finite group. The from checking small cases in GAP it seems that every irrep of $ G \wr S_2 $ is of the form $ V_i \otimes V_i $ for some irrep of $ G $ or of the form $ (V_i \otimes V_j) \oplus (V_j \otimes V_i) $ for a pair of distinct irreps $ V_i,V_j $ of $ G $.

What is the correct way to generalize this to $ G \wr S_n $?

It seems that if the degrees of the $ G $ irreps are multiset $ \{ d_i \} $ then every irrep of $ G \wr S_2 $ has degree $ d_i^2 $ or $ 2d_id_j $ where $ d_i, d_j $ are the degree of the distinct irreps $ V_i,V_j $.

In general if $ p $ is prime then I think every irrep of $ G \wr S_p $ has degree $ d_i^p $ or $$ p \prod_{\nu=1}^p d_{i_\nu} $$ where the $ d_{i_\nu} $ are the degrees of the distinct irreps $ V_{d_{i_\nu}} $

Is this the right generalization? How does the structure of representations extend to $ G \wr S_n $ for general $ n $?

Ian Gershon Teixeira
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  • Probably you are familiar with this, but in case some reader isn't: The case of an abelian $G$ is discussed in the answers to this question. May be more generally elsewhere on our site, but that's the one I recalled. Anyway, I think the mechanism of inducting from $G^n$ and considering the stabilizers is a starting point. – Jyrki Lahtonen Jun 07 '25 at 02:43
  • So for example, if $n=5$, $G$ is cyclic of order $m>2$, $\psi$ is the character of $G$ that maps the generator of $G$ to $e^{2\pi i/m}$, and $\chi$ is the character of $G^5$ defined by $$\chi:(g_1,g_2,g_3,g_4,g_5)\mapsto \psi(g_1g_2g_3g_4^{-1}g_5^{-1}),$$ then $\chi$ extends to a character of $H:=(G^5)\rtimes(Sym({1,2,3})\times Sym({4,5})$. And, at the last step inducing from $H$ to $G\wr S_5$ yields a 10-dimensional irreducible rep. Here $10$ is the index of the stabilizer. – Jyrki Lahtonen Jun 07 '25 at 04:09
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    In other words, I think it is more complicated than your hunch might indicate. – Jyrki Lahtonen Jun 07 '25 at 04:11
  • I dare guess this thread discusses a generalization of my example. – Jyrki Lahtonen Jun 07 '25 at 04:57
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    Your guess can't be right because it doesn't reproduce the right degrees when $G$ is trivial; you should be getting the degrees of the irreps of $S_p$ which are, of course, complicated (given by the hook length formula). – Qiaochu Yuan Jun 07 '25 at 13:22
  • The correct answer is here: https://ncatlab.org/nlab/show/wreath+product+of+groups#IrreducibleRepresentations – Qiaochu Yuan Jun 07 '25 at 13:42

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