  In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which
are invariant under the action of finite reduction groups. We present
6-dimensional YB maps corresponding to Darboux transformations for the
Nonlinear Schr\"odinger (NLS) equation and the derivative Nonlinear
Schr\"odinger (DNLS) equation. These YB maps can be restricted to
$4-$dimensional YB maps on invariant leaves. The former are completely
integrable and they also have applications to a recent theory of maps
preserving functions with symmetries \cite{Allan-Pavlos}. We give a $6-$
dimensional YB-map corresponding to the Darboux transformation for a
deformation of the DNLS equation. We also consider vector generalisations of
the YB maps corresponding to the NLS and DNLS equation.
