From Kronecker to tableau pseudo-characters in tensor models

We present a brief summary of the recent discovery of direct tensorial analogue of characters. We distinguish three degrees of generalization: (1) c-number Kronecker characters made with the help of symmetric group characters and inheriting most of the nice properties of conventional Schur functions, except for forming a complete basis for the case of rank r > 2 tensors: they are orthogonal, are eigenfunctions of appropriate cut-and-join operators and form a complete basis for the operators with non-zero Gaussian averages; (2) genuine matrix-valued tensorial quantities, forming an over-complete basis but difficult to deal with; and (3) intermediate tableau pseudo-characters, depending on Young tables rather than on just Young diagrams, in the Kronecker case, and on entire representation matrices, in the genuine one. (C) 2018 The Authors. Published by Elsevier B.V.