Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras

Abstract

We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients (Clebsch-Gordan multiplicities of symmetric groups). The color-exchange symmetry present in the Gaussian model, as well as a large class of interacting models, is used to refine the description of the permutation centralizer algebras. This discussion is extended to a general number of colors $d$: it is used to prove the integrality of an infinite family of number sequences related to color-symmetrizations of colored graphs, and expressible in terms of symmetric group representation theory data. Generalizing a connection between matrix models and Belyi maps, correlators in Gaussian tensor models are interpreted in terms of covers of singular 2-complexes. There is an intriguing difference, between matrix and higher rank tensor models, in the computational complexity of superficially comparable correlators of observables parametrized by Young diagrams.

81 pages; 5 figures; 5 tables; references updated and added

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• SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry
• Invariant Operators, Orthogonal Bases and Correlators in General Tensor Models
• Tensor and Matrix models: a one-night stand or a lifetime romance?
• BPS states, conserved charges and centres of symmetric group algebras
• Tensorial generalization of characters
• W-representation of Rainbow tensor model
• The quantum detection of projectors in finite-dimensional algebras and holography
• Spontaneous Symmetry Breaking in Tensor Theories
• Generalized cut operation associated with higher order variation in tensor models
• Permutation invariant Gaussian 2-matrix models
• All-orders asymptotics of tensor model observables from symmetries of restricted partitions
• A Renormalizable SYK-type Tensor Field Theory
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• Renormalizable Enhanced Tensor Field Theory: The quartic melonic case
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• Matrix and tensor witnesses of hidden symmetry algebras
• Kronecker coefficients from algebras of bi-partite ribbon graphs
• Backgrounds from Tensor Models: A Proposal
• Combinatorics of KP hierarchy structural constants
• (Anti-)Symmetrizing Wave Functions
• Dyonic Objects and Tensor Network Representation