The Integral Tree Representation of the Symmetric Group
نویسنده
چکیده
Let Tn be the space of fully-grown n-trees and let Vn and V ′ n be the representations of the symmetric groups n and n+1 respectively on the unique non-vanishing reduced integral homology group of this space. Starting from combinatorial descriptions of Vn and V ′ n , we establish a short exact sequence of Z n+1-modules, giving a description of V ′ n 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 n , Lien . Therefore we have an explicit combinatorial description of the integral representation of n+1 on Lien and this representation fits into a short exact sequence involving Lien and Lien+1.
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