= Tensor-square decomposition of the defining sp4 representation
For the defining four-dimensional representation $V$ of the <symplectic Lie algebra> $\mathfrak{sp}_4$, with the short <simple root> numbered first, its <highest weight> is $\omega_1$. Its <tensor square> decomposes as
$$
V\otimes V\cong V(2\omega_1)\oplus V(\omega_2)\oplus V(0),
$$
with dimensions $10,5,1$. The first summand is the <symmetric square>; the other two form the <exterior square>, split by <symplectic contraction of an exterior square>.
Back to article page
