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

## TL;DR

This paper identifies a small, efficient set of invariants that distinguish two copies of the natural $S_n$-representation, providing minimal separating sets for small n, advancing understanding of invariant rings.

## Contribution

It introduces a significantly smaller set of separating invariants for the ring of vector invariants of two copies of the natural $S_n$-representation, with minimality proven for small n.

## Key findings

- Set of separating invariants is much smaller than generating sets.
- For n ≤ 4, the set is minimal among all separating sets.
- Provides explicit construction of separating invariants.

## Abstract

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.