On representation matrices of boundary conditions in SU(n) gauge theories compactified on two-dimensional orbifolds

We study the existence of diagonal representatives in each equivalence class representation matrices boundary conditions $SU(n)$ or $U(n)$ gauge theories compactified on orbifolds $T^2/{\mathbb Z}_N$ ($N = 2, 3, 4, 6$). suppose that theory has a global $G' U(n)$ symmetry. Using constraints, unitary transformations and transformations, we examine whether can simultaneously become not. show at least one representative necessarily exists Z}_2$ Z}_3$, but Z}_4$ Z}_6$ contain not only also non-diagonal $2 \times 2$ ones $3 3$ ones, respectively, as members block-diagonal submatrices. These have discrete parameters, which means rank-reducing symmetry breaking be caused by Wilson line phases.