Past exam of the groups and representations course of the University of Oxford / 2025 / 4 / a

For the A5 root system, take simple roots ${\alpha }_i=e_i-e_{i+1}$, $i=1,\dots ,5$, in the hyperplane ${\sum }_{i=1}^6{x}_i=0$. Their squared lengths are two, adjacent simple roots have inner product $-1$, and all other pairs are orthogonal. Hence the An Dynkin diagram is the five-node single-edge chainIts Cartan matrix isThe complex Lie algebra is ${\mathfrak{sl}}_6(\mathbb{C})$; the corresponding compact special unitary group is $SU(6)$, whose Lie algebra is its compact real form.