Nested set complexes of Dowling lattices and complexes of Dowling trees

Details Nested set complexes of Dowling lattices and complexes of Dowling trees Nested set complexes of Dowling lattices and complexes of Dowling trees

2006-03-15

arxiv.org/abs/math/0603383v2

Mathematics

Combinatorics

14 pages, 2 figures, corrected typos, added references

Scientific paper

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).

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