Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras

TL;DR

This paper explores the algebraic structure of tensor models using permutation centralizer algebras, revealing their semi-simplicity, connections to symmetric group representation theory, and implications for correlator computations.

Contribution

It introduces a new algebraic framework for tensor models based on permutation centralizer algebras, linking correlators to Kronecker coefficients and extending to multiple colors.

Findings

Permutation centralizer algebras are semi-simple and decomposed via Clebsch-Gordan coefficients.

Correlators are expressed in terms of Kronecker coefficients, connecting to symmetric group theory.

The approach proves integrality of certain sequences related to colored graph symmetrizations.

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.