# The Integral Tree Representation of the Symmetric Group

Research paper by Sarah Whitehouse

Indexed on: 01 May '01Published on: 01 May '01Published in: Journal of Algebraic Combinatorics

#### Abstract

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 $$\mathbb{Z}\Sigma _{n + 1}$$-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.Modulo a twist by the sign representation, Vn is shown to be dual to the Lie representation of Σn, Lien. Therefore we have an explicit combinatorial description of the integral representation of Σn+1 on Lien and this representation fits into a short exact sequence involving Lien and Lien+1.