2017/08/31 by Joseph Ben Geloun, Sanjaye Ramgoolam · 40 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Basis (linear algebra) #Centralizer and normalizer #Kronecker delta #Matrix (chemical analysis) #Noncommutative and Quantum Gravity Theories #Observable #Permutation (music) #Rank (graph theory) #Symmetric group #Symmetric tensor #Tensor (intrinsic definition) #hep-th #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/jhep11(2017)092
published in Journal of High Energy Physics 2017(11) (Springer Nature) · 81 pages; 5 figures; 5 tables; references updated and added
openalex created_date 2017/08/31 · arxiv created 2017/09/11 · openalex publication_date 2017/11/01 · arxiv updated 2020/04/27 · openalex updated_date 2026/08/05
A bstract 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.