Symmetric Khovanov--Rozansky link homologies

We provide a finite-dimensional categorification of the symmetric evaluation of sl(N)-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of sl(N). The construction is made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov--Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.