I am a physicist, so my apologies if haven't framed the question in the proper mathematical sense. Matrices are used as group representations. Matrices act on vectors. So in physics we use matrices to transform vectors and also to denote the symmetries of the vector space. v_i = Sum M_ij v_j Is there an equivalent group representation that act on tensors instead of vectors? T_ij = Sum M_iajb T_ab I guess then we need some group multiplication law... may be M_abcd = Sum M_axcy M_xbyd Does this make any sense? Please provide some references where I can find more information.