Mackey's test for irreducibility of induced representation over $\mathbb{C}$ is:

Let $G$ be a finite group, $H\leq G$, $W$ be a representation of $H$, and $W^x$ be conjugate representation of $H^x=xHx^{-1}$. Then following are equivalent:

(i) $Ind^G_H(W)$ is irreducible.

(ii) $W$ is irreducible and for each $x\in G\setminus H$, $W$ and $W^x$ have no common irreducible component, when restricted to $H\cap H^x$.

There is question posted, about changing the ground field, and the answer posted is (are) "Yes".

But, $(ii)\Rightarrow (i)$ is true for any field of characteristic zero or prime to $|G|$; how does $(i)\Rightarrow (ii)$ for such fields?


No. Over $\mathbb R$ let $G$ be the quaternion group of order $8$, $H$ the subgroup of order $2$, $W$ the nontrivial one-dimensional representation.

EDIT For an even simpler example, see Kevin Ventullo's comment!

  • 2
    $\begingroup$ Even simpler: induce the nontrivial character of $Z/2$ up to $Z/4$ over $\mathbb{R}$. $\endgroup$ – Kevin Ventullo Jan 11 '12 at 16:56
  • $\begingroup$ Duh. Of course. You should write that as an answer. $\endgroup$ – Tom Goodwillie Jan 11 '12 at 17:55
  • $\begingroup$ Well, I figure people can just read the comment. For some reason, I like it better when there's only one answer. $\endgroup$ – Kevin Ventullo Jan 12 '12 at 20:06

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