  We discuss the use of group symmetries in optimisation, in particular with
respect to the structure of subdifferential and projection operators. This
allows us to generalise a classic result of Adrian Lewis regarding the
characterisation of the subdifferential of a permutation invariant convex
function to the characterisation of the proximal subdifferential of a Schur
convex function that is invariant with respect to a finite reflection group. We
are also able to simplify and generalise results on projections onto symmetric
sets, in particular, we study projections on sparsity constraints used in
sparse signal recovery and compressed sensing.
