  The controversy regarding the ferroelectric behavior of hexagonal InMnO$_3$
is resolved by using a combination of x-ray diffraction (XRD), piezoresponse
force microscopy (PFM), second harmonic generation (SHG), and density
functional theory (DFT). While XRD data show a symmetry-lowering unit-cell
tripling, which is also found in the multiferroic hexagonal manganites of
$P6_3cm$ symmetry, PFM and SHG do not detect ferroelectricity at ambient or low
temperature, in striking contrast to the behavior in the multiferroic
counterparts. We propose instead a centrosymmetric $P\bar{3}c$ phase as the
ground state structure. Our DFT calculations reveal that the relative energy of
the ferroelectric and nonferroelectric structures is determined by a
competition between electrostatics and oxygen-$R$-site covalency, with an
absence of covalency favoring the ferroelectric phase.
