The genus series for maps
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.
scientific article; zbMATH DE number 858624
scientific article