From Kronecker to tableau pseudo-characters in tensor models

TL;DR

This paper explores the development of tensorial analogues of characters, introducing tableau pseudo-characters as an intermediate form that bridges simpler Kronecker characters and more complex matrix-valued tensor quantities.

Contribution

It introduces tableau pseudo-characters for tensors, providing a new intermediate framework that extends Kronecker characters to more complex tensor models.

Findings

Kronecker characters inherit many properties of Schur functions.

Genuine tensorial quantities form an over-complete basis.

Tableau pseudo-characters depend on Young tables and matrices.

Abstract

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.