On nonnegative invariant quartics in type A

The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse $A_n$-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.