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Let $G$ be a finite group and we take for the field the complex numbers. Call an irreducible character $\xi$ with degree $m$ of $G$ perfect, if all exterior powers $\bigwedge\nolimits^k \xi$ are irreducible for $k=1,\dotsc,m$.

Question 1: Can perfect irreducible characters be characterised in an easy way?

Question 2: What are the perfect irreducible characters for the symmetric group?

A famous example is the standard representation, but there are more. For example $S_6$ has 6 such irreducible perfect characters.

Question 3: What are the finite groups having a faithful irreducible character that is perfect? For $2$-groups the number of such groups seems rather small and starts for 2-groups with $n \geq 2$ elements with 1,1,3,6,9,12,15.

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    $\begingroup$ The standard representation of $S_n$ is not irreducible. I think you mean the reflection representation, with character $\chi^{(n-1,1)}$. $\endgroup$ Apr 17 at 18:42
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    $\begingroup$ @RichardStanley This is called the standard representation in some books, for example the one of Lorenz. It is the representation $V$ such that $M \cong V \oplus K$, where $M$ is the natural representation of the symmetric group given by the operation on the set $\{1,...,n\}$. $\endgroup$
    – Mare
    Apr 17 at 18:46

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For an irreducible of degree $d$ to have this property, the sum of the squares of the $\binom{d}{i}$ with $2i\leqslant d$ has to be at most the group order. Now look at faithful irreducibles of $S_n$ that are not associated to $(n-1,1)$ or $(2,1^{n-2})$. For $n=7$ and $n=8$ the smallest have dimension $14$. If $n\geqslant 9$, the smallest are $(n-2,2)$ and $(2^2,1^{n-4})$, with dimension $n(n-3)/2$. I'm pretty sure this is already too big for the sum of the squares to be at most $n!$. See Sloane A277247.

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  • $\begingroup$ This is supposed to be an answer to the second of your two (now three) questions. $\endgroup$ Apr 17 at 19:29
  • $\begingroup$ The upshot is that the dimension has to be very small compared with the group order, so it will be a rare phenomenon. $\endgroup$ Apr 17 at 19:39
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    $\begingroup$ For $d$ odd, $\sum_{i=0}^{\lfloor d/2} \binom{d}{i}^2$ is $\binom{2d-1}{d-1}$. For $d$ even, it is $(1/2)(\binom{2d}{d} + \binom{d}{d/2}^2)$. Either way, roughly $4^d$. $\endgroup$ Apr 17 at 19:41
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    $\begingroup$ It looks like $S_n \ltimes C^n$, for $C$ any cyclic group, might be another example. And some small variants on this: $S_n \ltimes C^{n-1}$, $A_n \ltimes C^n$ and $A_n \ltimes C^{n-1}$. $\endgroup$ Apr 17 at 19:44
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    $\begingroup$ Indeed, for any complex reflection group, the reflection representation has this property. See the answer at math.stackexchange.com/questions/3382199/… $\endgroup$ Apr 17 at 20:23

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