# Characters with all higher exterior powers irreducible

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.

• The standard representation of $S_n$ is not irreducible. I think you mean the reflection representation, with character $\chi^{(n-1,1)}$. Apr 17 at 18:42
• @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 $\{1,...,n\}$.
– Mare
Apr 17 at 18:46

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.
• 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 $4^d$. Apr 17 at 19:41
• 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 $A_n \ltimes C^{n-1}$. Apr 17 at 19:44