Nested set complexes of Dowling lattices and complexes of Dowling trees
Abstract
Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice Q(G) and of its subposet of the G-symmetric partitions Q_G which was recently introduced by Hultman together with the complex of G-symmetric phylogenetic trees T_G. Hultman shows that T_G and Q_G are homotopy equivalent and Cohen-Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov shows that in fact T_G is subdivided by the order complex of Q_G. We introduce the complex of Dowling trees T(G) and prove that it is subdivided by the order complex of Q(G) and contains T_G as a subcomplex. We show that T(G) is obtained from T_G by successive coning over certain subcomplexes. We explicitly and independently calculate how many homology spheres are added in passing from T_G to T(G).
Keywords
Cite
@article{arxiv.math/0603383,
title = {Nested set complexes of Dowling lattices and complexes of Dowling trees},
author = {Emanuele Delucchi},
journal= {arXiv preprint arXiv:math/0603383},
year = {2007}
}
Comments
14 pages, 2 figures, corrected typos, added references