B2 root system ($B_2$)

The $B_2$ root system consists of $\pm {\epsilon }_1$, $\pm {\epsilon }_2$, and $\pm {\epsilon }_1\pm {\epsilon }_2$. Its vector representation has weights ${\epsilon }_1,{\epsilon }_2,0,-{\epsilon }_2,-{\epsilon }_1$.

For simple roots $(1,0),(-1,1)$, Dynkin labels of a Euclidean weight $(x,y)$ are $(2x,-x+y)$. The highest root $(1,1)$ gives the Adjoint representation, of dimension ten. Its eight nonzero weights are the eight roots of $B_2$; the zero weight has multiplicity two. The nine distinct label pairs are $(2,0),(0,1),(-2,2),(2,-1),(0,0),(-2,1),(2,-2),(0,-1),(-2,0)$.

For simple roots $(1,0)$ and $(-1,1)$, coroot integrality gives the weight lattice $P={\mathbb{Z}}^2\cup (\mathbb{Z}+1/2{)}^2$. This is the lattice for the Spin group $\mathrm{Spin}(5)$ and its Lie algebra. Representations descending to $SO(5)$ use the integer coset, because a $2\pi$ rotation in either coordinate plane must act trivially.

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.

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