Matter Representations and Gauge Symmetry Breaking via Compactified Space

We study dynamical gauge symmetry breaking via compactified space in the framework of SU($N$) gauge theory on $M^{d-1}\times S^1$ ($d=4,5,6$) space-time. Especially, we study in detail the gauge symmetry breaking in SU(2) and SU(3) gauge theories when the models contain both fundamental and adjoint matters. As the result, we find that any pattern of gauge symmetry breaking is realized by selecting appropriate set of numbers $(\Nf,\Nad)$ in these cases. It is achieved without tuning boundary conditions of matter fields. As the by-product, in some cases we get effective potential which has no curvature at the minimum thus leading to massless Higgs scalars, irrespectively of the size of compactified space.