Observables in a tensor model can be enumerated using colored graphs, which have a description in terms of permutation triples subject to an equivalence relation generated by permutation products. A gauge-fixing of the equivalence relation relates the enumeration to bipartite ribbon graphs and Belyi maps between surfaces. Fourier transformation on permutation group algebras relates the counting of observables to a sum of squares of Kronecker coefficients. In the recent paper https://arxiv.org/abs/2010.04054, we develop these observations to give a combinatoric construction of the Kronecker coefficient for a fixed triple of Young diagrams. The Kronecker coefficient counts vectors in a lattice of ribbon graphs, determined as null vectors of integer matrices. The result motivates a discussion of quantum mechanical systems and algorithms to determine non-vanishing Kronecker coefficients. Using the link to Belyi maps, these quantum mechanical systems have an interpretation in terms of quantum membrane geometries interpolating between algebraic string worldsheets.
Joseph Ben Geloun
Fabien Vignes-Tourneret
