Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 5 / c

Realize the B2 root system in ${\mathbb{R}}^2$ asChoose the short simple root and long simple rootThenas follows from $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$. The roots form a square from the long roots with the four short roots on the coordinate axes; the double edge in the Dynkin diagram points toward ${\alpha }_1$.

The positive roots areFor $\lambda =a{\omega }_1+b{\omega }_2$, substituting their coroot pairings in the Weyl dimension formula givesThis is the Weyl dimension formula for B2.

The weights of the defining five-dimensional representation areIts highest weight is ${\epsilon }_1={\omega }_2$, so irreducibility identifies it as

If $v_{+}$ is a highest-weight vector, then $v_{+}\otimes v_{+}$ is a highest-weight vector of weight $2{\omega }_2$ in $V\otimes V$. The subrepresentation it generates is therefore $V(2{\omega }_2)$. Equivalently, it is the Traceless symmetric square of the defining so5 representation; the invariant quadratic form supplies the complementary trivial line in $S^{2}V$.

Putting $a=0$ and $b=2$ into the formula from part i gives