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The symmetry conditions for the development of a macroscopic electricalpolarization as a secondary order parameter to a magnetic ordering transition,and the constraints on the direction of the polarization vector, are determinedby a non-conventional application of the theory of irreducibleco-representations. In our approach, which is suitable for both magnetic andstructural modulations, anti-unitary operators are employed to describesymmetry operations that exchange the propagation vector $\textbf{k}$ with$\textbf{-k}$, rather than operations combined with time-reversal as inclassical \textit{corep} analysis. Unlike the conventional irreduciblerepresentations, co-representations can capture the full symmetry properties ofthe system even if the propagation vector is in the interior of the Brillouinzone. It is shown that ferroelectricity can develop even for a completelycollinear structure, and that helical and cycloidal magnetic structures are notalways polar. In some cases, symmetry allows the development of polarizationparallel to the magnetic propagation vector. Our analysis also highlights theunique importance of magnetic commensurability, enabling one to derive thedifferent symmetry properties of equivalent commensurate and incommensuratephases even for a completely generic propagation vector

Symmetry constraints on the electrical polarization in multiferroic materials