Counting Distinct Plaquette Phase Configurations under Dihedral Symmetry

We consider a square plaquette whose four directed links carry distinct quantized phases (a, b, c, d). When rotations and reflections of the plaquette are regarded as symmetry operations, two assignments related by such a symmetry are considered equivalent. By applying Burnside’s lemma to the dihedral group D4, we computethe number of inequivalent configurations. The result shows that there exist precisely three distinct configurations, consistent with the classical formula (n − 1)!/2 for n = 4. This simple enumeration highlights the combinatorial structure of discrete gauge-phase arrangements on a symmetric lattice plaquette.