We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in [1].These maps satisfy an algebraic property which makes it easyto study their positivity ork-positivity.They are therefore particularly suitable for applicationsto entanglement detection in quantum informationtheory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call(a, b)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitaryconjugation as studied by Bhat in [2] and as a generalizationof the equivariant maps studied in [1].Using representation theory, we fully compute them and study their graphical representation, and showthat they are basically enough to study all equivariant maps. We finally apply them to the problem ofentanglement detection and prove that they form a sufficient (infinite) family of positive maps to detectallk-entangled density matrices.