Weyl constructions for finite groups

Question

Let $G$ be a finite group.

Is there a complex finite dimensional irreducible representation $V$ such that all irreducible ones are submodules of $V^{\otimes n}$ for some $n \in \mathbb{N}$?

If not, what's the smallest group $G$ for this to fail?

Answer 1

It's a nice exercise in character theory to show that for $G$ a finite group and $V$ a representation, every irreducible representation occurs in $V^{\otimes n}$ for some $n$ iff $V$ is faithful (faithfulness is obviously necessary and sufficiency is the interesting direction).

So your question is equivalent to asking when $G$ has a faithful irreducible representation. A necessary condition, by Schur's lemma, is that the center $Z(G)$ must be cyclic (since it embeds into $\mathbb{C}^{\times}$), so $C_2 \times C_2$ (the smallest non-cyclic group) is the smallest counterexample.

Edit: Regarding the nice exercise above, here is the argument I had in mind.

Proposition: Let $G$ be a finite group and $V$ a faithful finite-dimensional representation of it. Then every irreducible representation occurs in $V^{\otimes n}$ for some $n$.

Proof. Let $W$ be some irreducible and consider the characters $\chi_W, \chi_V$. We want to show that

$$\langle \chi_W, \chi_V^n \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi_W}(g) \chi_V(g)^n$$

is nonzero for some $n$. The idea of this proof is to consider what happens to this sum when $n$ is large: it's dominated by the terms where $|\chi_V(g)|$ is largest and all the other terms exponentially decay in comparison. Using the fact that $\chi_V(g)$ is the sum of the eigenvalues of $\rho_V(g)$, which are roots of unity, $|\chi_V(g)|$ has maximum value $\dim V$ and takes this maximum value iff all the eigenvalues of $\rho_V(g)$ are the same iff $\rho_V(g)$ is a scalar multiple of the identity (and the scalar must be some root of unity).

In the simplest case the only $g$ with this property is the identity; this happens, for example, in David Speyer's answer explaining a version of this proof in response to this MO question because he added a copy of the trivial representation. In that case we get that the sum above is asymptotically $\frac{(\dim V)^n}{|G|} + O(c^n)$ where $|c| < \dim V$ and hence must be eventually positive, and with a little more effort we could give an upper bound on the smallest $n$ at which this occurs.

Instead of doing that we can just restrict our attention to the values of $n$ such that if $\rho_V(g)$ is a scalar multiple of the identity then $\rho_V(g)^n$ is the identity (at worst this means restricting our attention to $|G| \mid n$). If there are $k$ different elements of $g$ acting by a scalar then we get, for these restricted values of $n$, that the sum above is asymptotically $k \frac{(\dim V)^n}{|G|} + O(c^n)$ where $|c| < \dim V$, and again the conclusion follows. $\Box$

Less analytic and more algebraic arguments are possible that would generalize, for example, to fields of characteristic $0$. This argument has the benefit of generalizing, with some modification, to the case that $G$ is a compact group (although we need to consider irreducibles in $V^{\otimes n} \otimes (V^{\ast})^{\otimes m}$ in this case), as I learned from David Speyer on MO here.