Abstract
Let T n be the space of fully-grown n-trees and let V n and V n ′ be the representations of the symmetric groups Σ n and Σ n+1 respectively on the unique non-vanishing reduced integral homology group of this space. Starting from combinatorial descriptions of V n and V n ′, we establish a short exact sequence of \(\mathbb{Z}\Sigma _{n + 1} \)-modules, giving a description of V n ′ in terms of V n and V n+1. This short exact sequence may also be deduced from work of Sundaram.
Modulo a twist by the sign representation, V n is shown to be dual to the Lie representation of Σ n , Lie n . Therefore we have an explicit combinatorial description of the integral representation of Σ n+1 on Lie n and this representation fits into a short exact sequence involving Lie n and Lie n+1.
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Whitehouse, S. The Integral Tree Representation of the Symmetric Group. Journal of Algebraic Combinatorics 13, 317–326 (2001). https://doi.org/10.1023/A:1011264315849
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DOI: https://doi.org/10.1023/A:1011264315849