From Kronecker to tableau pseudo-characters in tensor models

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.

9 pages

References in corpus (17)

• Uncolored Random Tensors, Melon Diagrams, and the SYK Models
• A Generalization of Sachdev-Ye-Kitaev
• Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
• Random tensor models in the large N limit: Uncoloring the colored tensor models
• The complete $1/N$ expansion of a SYK--like tensor model
• All point correlation functions in SYK
• EPRL/FK Group Field Theory
• Lost in Translation: Topological Singularities in Group Field Theory
• On the complete perturbative solution of one-matrix models
• Correlators in tensor models from character calculus
• SYK Models and SYK-like Tensor Models with Global Symmetry
• Correlators in the $\mathcal{N}=2$ Supersymmetric SYK Model
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• Supersymmetric SYK Model: Bi-local Collective Superfield/Supermatrix Formulation
• Introduction to Non-Linear Algebra
• Bi-Local Holography in the SYK Model: Perturbations

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• Cauchy formula and the character ring
• Tensorial generalization of characters
• Kerov functions for composite representations and Macdonald ideal
• Quantization of Harer-Zagier formulas
• W-representation of Rainbow tensor model
• Spontaneous Symmetry Breaking in Tensor Theories
• W-representations of the fermionic matrix and Aristotelian tensor models
• Generalized cut operation associated with higher order variation in tensor models
• On generalized Macdonald polynomials
• Holography for Tensor models
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Hook variables: cut-and-join operators and $\tau$-functions
• Backgrounds from Tensor Models: A Proposal
• Notes about KP/BKP correspondence
• The Non-melonic Sector of Tensor Models and Gravity
• Review on the Operator/Feynman diagram/Dessins d'enfant Correspondence in Tensor Model