Separating Invariants for Two Copies of the Natural $S_n$-Action

This note provides a set of separating invariants for the ring of vector invariants $K[V^2]^{S_n}$ of two copies of the natural $S_n$-representation $V = K^n$ over a field of characteristic 0. This set is much smaller than generating sets of $K[V^2]^{S_n}$. For $n \leq 4$ we show that this set is minimal with respect to inclusion among all separating sets.