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

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.

21 pages, 7 figures

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Cited by in corpus (5)

• Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$
• SU(N) polynomial integrals and some applications
• Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$
• Commutators of random matrices from the unitary and orthogonal groups
• Random stochastic matrices from classical compact Lie groups and symmetric spaces