Absence of twisting for non-trivial discrete torsion

We study discrete torsion for the $n$--torus with finite symmetry group $G$ from the Dijkgraaf--Witten viewpoint. A class in $H^n(G,U(1))$ assigns a phase to each flat $G$--bundle, equivalently to each commuting $n$--tuple in $G$ up to conjugation. We introduce the subgroup $\Br^n(G)\subseteq H^n(G,U(1))$ of \emph{untwisted} classes, those whose Dijkgraaf--Witten phases are trivial on all commuting tuples, and derive a universal coefficient exact sequence involving this invariant. In degree $2$ this recovers the Bogomolov multiplier / unramified Brauer group. We implement algorithms for computing $\Br^n(G)$ and corresponding torus partition functions, and report on computations for families of finite subgroups of $\SU(4)$.