I have a question about representation theory of finite group $G$ (you may assume that the field has char 0). If you have a morphism of representations $f\colon V\to W$ then you can restrict it on irreducible components of these representations and get (by the Schur's lemma) the set of scalar mappings $f\mid_{V_i}\colon V_i\to W_i$ (scalars determined unique up to the change of bases in $\Bbbk^n\otimes V_i$). At least we can know for each $i$ the sum of these scalars. Is there any way to compute or discover something about these scalars or this sum? I think the answer may be given in terms of some sort of scalar product of characters, but I have no direct link for this (I've read the Serre's book on representation theory and haven't find an answer).

This question comes from the topology. If you want to compute the functorial higher Massey coproducts on simplicial complexes you'll have to compute it first on simplex $\Delta^{m-1}$ (it means the computing the sequence of linear mappings $\delta_k\colon\bigoplus_{i=0}^m \Lambda^i(\Bbbk^m)\to(\bigoplus_{i=0}^m \Lambda^i(\Bbbk^m))^{\otimes k}$ with some relations between them). These Massey coproducts seem to be $S_m$-equivariant morphisms. If you want to get the Massey coproducts invariant under the baricentric subdivision $B\Delta^{m-1}$ then there are endomorphisms of the spaces of $\delta_k\colon\bigoplus_{i=0}^m \Lambda^i(\Bbbk^m)\to(\bigoplus_{i=0}^m \Lambda^i(\Bbbk^m))^{\otimes k}$ and you have to compute the eigenvalues and eigenvectors of such endomorphisms (I'm not giving any detailed motivation - but there is an article about it - I can give the link). So I'm just finding the the way to compute these eigenvalues and eigenvectors.