The genus series for maps
type: Item
instance of: scholarly article
title: The genus series for maps
zbMATH Open document ID: 0844.05040
DOI: 10.1016/0022-4049(94)00155-3
published in: Journal of Pure and Applied Algebra
publication date: 1996-08-22T00:00:00Z
review text: A rooted map is a 2-cell embedding of a connected unlabelled graph, with loops and multiple edges allowed, in a closed orientable surface, where a mutually incident vertex, edge, and face is distinguished. Two rooted maps are equivalent, if one is mapped to the other by an orientation preserving homeomorphism of the surface, keeping the distinguished objects. The genus series is the generating series for inequivalent rooted maps with genus \(g\), \(i_r\) vertices of degree \(r\) \((r= 1, 2,\dots)\), \(k\) edges, \(j\) faces. An integral representation for the genus series has been obtained by \textit{D. Bessis}, \textit{C. Itzykson} and \textit{J. B. Zuber} [Adv. Appl. Math. 1, 109-157 (1980; Zbl 0453.05035)], and \textit{G. 't Hooft} [Nucl. Phys. B 72, 461-473 (1974)] using properties of the Haar measure for the unitary group and Weyl integration theorems. The present paper gives a finitary derivation of the integral representation of the genus series, apart from the existence \(\int_R e^{- x^2} dx\). This is based on a combinatorial representation of embeddings (rotation scheme), and on properties of the centre of the group algebra of the symmetric group.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 57N37
zbMATH DE Number: 858624
zbMATH Keywords: symmetric functions
zbMATH Keywords: connected graph
zbMATH Keywords: rotation scheme
zbMATH Keywords: rooted map
zbMATH Keywords: embedding
zbMATH Keywords: surface
zbMATH Keywords: genus series
zbMATH Keywords: integral representation
zbMATH Keywords: Haar measure
zbMATH Keywords: unitary group
zbMATH Keywords: Weyl integration theorems
zbMATH Keywords: combinatorial representation
zbMATH Keywords: symmetric group
author: David M. Jackson
reviewed by: László A. Székely
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00155-3
OpenAlex ID: W2013844898
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description: scientific article; zbMATH DE number 858624