Fundamental representations of B2 ($[1,0],[0,1]$)

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are ${\omega }_1=(1/2,1/2)$ and ${\omega }_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm 1/2,\pm 1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm 1,0),(0,\pm 1),(0,0)$.

When ${\alpha }_1$ is the short simple root, ${\omega }_1$ is the spin fundamental weight and ${\omega }_2$ is the vector fundamental weight. The Weyl dimension formula gives

For the five-dimensional defining representation $V=V({\omega }_2)$ of ${\mathfrak{so}}_5$, the invariant quadratic form splitsThus the traceless symmetric square is irreducible of dimension $15-1=14$.

For the five-dimensional defining representation of ${\mathfrak{so}}_5$, the symmetric square is the sum of its invariant trace and the traceless representation $V(2{\omega }_2)$, while the exterior square is the Adjoint representation $V(2{\omega }_1)$. Hence