Kronecker coefficients from algebras of bi-partite ribbon graphs

TL;DR

This paper explores the algebra of bi-partite ribbon graphs and its connection to Kronecker coefficients, providing new combinatorial tools for their computation in the context of matrix and tensor models.

Contribution

It introduces the algebra al{K}(n) of bi-partite ribbon graphs and links it to symmetric group algebras and Kronecker coefficients, offering novel combinatorial algorithms.

Findings

The algebra al{K}(n) is related to symmetric group algebras.

A matrix-block decomposition of al{K}(n) involves Kronecker coefficients.

Quantum models based on al{K}(n) enable combinatorial algorithms for Kronecker coefficients.

Abstract

Bi-partite ribbon graphs arise in organising the large $N$ expansion of correlators in random matrix models and in the enumeration of observables in random tensor models. There is an algebra $\mathcal{K}(n)$, with basis given by bi-partite ribbon graphs with $n$ edges, which is useful in the applications to matrix and tensor models. The algebra $\mathcal{K}(n)$ 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 $\mathcal{K}(n)$ as Hilbert spaces can be used to give combinatorial algorithms for computing the Kronecker coefficients.