The Integral Tree Representation of the Symmetric Group

Let Tn be the space of fully-grown n-trees and let Vn and Vn′ be the representations of the symmetric groups Σn and Σn+1 respectively on the unique non-vanishing reduced integral homology group of this space. Starting from combinatorial descriptions of Vn and Vn′, we establish a short exact sequence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}\Sigma _{n + 1} $$ \end{document}-modules, giving a description of Vn′ in terms of Vn and Vn+1. This short exact sequence may also be deduced from work of Sundaram.