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.
Type de document :
Article dans une revue
Journal of High Energy Physics, Springer, 2017, 11, pp.092. 〈10.1007/JHEP11(2017)092〉
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Soumis le : jeudi 21 décembre 2017 - 00:56:37
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Joseph Ben Geloun, Sanjaye Ramgoolam. Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras. Journal of High Energy Physics, Springer, 2017, 11, pp.092. 〈10.1007/JHEP11(2017)092〉. 〈hal-01669803〉



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