From Kronecker to tableau pseudo-characters in tensor models

• Main Authors: H. Itoyama, A. Mironov, A. Morozov
• Format: Article
• Language: English
• Published: Elsevier 2019-01-01
• Series: Physics Letters B
• Online Access: http://www.sciencedirect.com/science/article/pii/S0370269318308463

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.