Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

TL;DR

This paper introduces a diagrammatic method to express polynomial Lie group integrals as sums over surface maps, linking them to permutation factorizations and revealing new combinatorial structures.

Contribution

It provides a novel diagrammatic representation of Lie group integrals and connects them to permutation factorizations, expanding the mathematical understanding of these integrals.

Findings

Representation of integrals as sums over surface maps

Identification of specific conditions for maps involved

Formulation of new permutation factorization problems

Abstract

Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.