From Kronecker to tableau pseudo-characters in tensor models

H Itoyama,A Mironov,A Morozov

doi:10.1016/j.physletb.2018.11.008

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.

Highlights

We present a brief summary of the recent discovery of direct tensorial analogue of characters
As soon as the characters χR do not form a complete basis in the space of all gauge-invariant operators, there are non-vanishing gauge-invariant operators that are not representable as sum of characters, but their Gaussian averages are: O = R aRχR, but O = R aRχR
Symmetric groups with n > 4 are non-solvable, the Kroneker coefficients can exceed one, and the number of gauge-invariant operators exceeds the number of Kroneker characters (7)

Summary

Introduction

We present a brief summary of the recent discovery of direct tensorial analogue of characters. We define the Kronecker tensorial character, depending on the r-tuple of Young diagrams as a direct generalization of (3): χR(M, M ) := n1!

Results

Conclusion