Large $N$ limit of irreducible tensor models: $O(N)$ rank-$3$ tensors with mixed permutation symmetry

It has recently been proven that in rank three tensor models, the anti-symmetric and symmetric traceless sectors both support a large $N$ expansion dominated by melon diagrams [arXiv:1712.00249 [hep-th]]. We show how to extend these results to the last irreducible $O(N)$ tensor representation available in this context, which carries a two-dimensional representation of the symmetric group $S_3$. Along the way, we emphasize the role of the irreducibility condition: it prevents the generation of vector modes which are not compatible with the large $N$ scaling of the tensor interaction. This example supports the conjecture that a melonic large $N$ limit should exist more generally for higher rank tensor models, provided that they are appropriately restricted to an irreducible subspace.