  We calculate the contribution to the polarization of $\Lambda$ hyperons in
relativistic nuclear collisions at high energy from the decays of
$\Sigma^*(1385)$ and $\Sigma^0$, which are the predominant sources of $\Lambda$
production besides the primary component, as a function of the $\Lambda$
momentum. Particularly, we estimate the longitudinal component of the mean spin
vector as a function of the azimuthal angle in the transverse plane, assuming
that primary $\Sigma^*$ and $\Sigma^0$ polarization follow the predictions of
local thermodynamic equilibrium in a relativistic fluid. Provided that the
rapidity dependence around midrapidity of polarization is negligible, we find
that this component of the overall spin vector has a very similar pattern to
the primary one. Therefore, we conclude that the secondary decays cannot
account for the discrepancy in sign between experimental data and hydrodynamic
model predictions of the longitudinal polarization of $\Lambda$ hyperons
recently measured by the STAR experiment at RHIC.
