A note on immanant preservers

We study the maps that transform one immanant into another. No surjectivity or linearity is imposed; the rudiments of the former are weakly embedded into the functional equation via $d_\chi(\Phi(A)+\lambda\Phi(B))=d_{\chi'}(A+\lambda B)$. We show that this property alone implies that $\Phi$ is linear and bijective.