Kronecker coefficients from algebras of bi-partite ribbon graphs

Abstract

Bi-partite ribbon graphs arise in organizing the large N expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra K(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {K}(n)$$\\end{document}, with basis given by bi-partite ribbon graphs with n edges, which is useful in the applications to matrix and tensor models. The algebra K(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {K}(n)$$\\end{document} is closely related to symmetric group algebras and has a matrix-block decomposition related to Clebsch–Gordan multiplicities, also known as Kronecker coefficients, for symmetric group representations. Quantum mechanical models which use K(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {K}(n)$$\\end{document} as Hilbert spaces can be used to give combinatorial algorithms for computing the Kronecker coefficients.