Non-negative integer matrix representations of finite groups

Question

I wanted to know all the non-negative integer matrix (NIM) irreducible representations (irrep) of finite groups i.e. the homomorphisms $\varphi : G \to GL(n,\mathbb{Z}_{\geq 0})$. By irreducible NIM rep (or NIM irrep), I mean that we can't write it into a direct sum of smaller NIM reps.

Edit: This proof shows that $GL(n,\mathbb{Z}_{\geq 0}) = S_n$ which simplifies things a lot.

For context: I am a physics grad student and these representations are called NIM reps in the physics literature. These NIM reps are useful when we study the action of topological defect lines on boundary states of 2D CFTs.

I state the things I am able to prove in chronological order:

1. Every matrix $M \in GL(n,\mathbb{Z}_{\geq 0})$ such that $M^m = I$ for some $m\in\mathbb{Z}_{+}$ is a permutation matrix. (Proof) This implies that matrices in NIM reps of finite groups are permutation matrices.
2. For the cyclic group of order prime $\mathbb{Z}_p$, the only non-trivial NIM irrep is the regular representation of $\mathbb{Z}_p$. (Here, irreducible = not a direct sum of smaller NIM reps)
3. For a group $\mathbb{Z}_n$, for every factor $k$ of $n$, there is a NIM irrep of $\mathbb{Z}_n$ which is the regular representation of $\mathbb{Z}_k$. Also, these are the only NIM irreps of $\mathbb{Z}_n$.

I would show my work, but it is too long to write here (there may be a shorter version out there) although the steps are elementary. If there is some obvious mistake in the three claims above, I would be more than happy to correct myself.

For non-abelian finite groups, I don't know how to proceed and how hard the problem is.

My question: Is the classification of NIM irreps of abelian and non-abelian finite groups known already? Could you state the results if possible or could you refer me to some literature on this? Thanks.

Answer 1

Your first statement about permutation matrices completely solves the problem, or at least reduces it to something known: any such representation is a permutation representation, so corresponds to an action of $G$ on some finite set $X$, and the irreducible ones are exactly the transitive group actions. Up to isomorphism transitive group actions are classified by conjugacy classes of subgroups of $G$: given such a subgroup $H$ the corresponding transitive group action is the action on the quotient $G/H$.

This is a result you'd encounter in any first course on group theory. It straightforwardly implies the classification you give for finite cyclic groups, and with a little effort (using e.g. Goursat's lemma) you can get a classification for finite abelian groups.

Complicated nonabelian groups have many different conjugacy classes of subgroups so this is quite a complicated set of objects in general. I don't even have a guess as to the number of conjugacy classes of subgroups of $S_n$, for example.