Flip parity in the SU2 tensor square
= Flip parity in the SU2 tensor square
On the multiplicity-one summand $V_{2n-2j}\subset V_n\otimes V_n$, interchange of tensor factors acts by $(-1)^j$. A highest-weight vector exhibiting the sign is
$$
\sum_{k=0}^j(-1)^k\binom jk
x^{n-k}y^k\otimes x^{n-j+k}y^{j-k}.
$$