Nested set complexes of Dowling lattices and complexes of Dowling trees

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 Qn(G) (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions Qn(G) which was recently introduced by Hultman (http://www.math.kth.se/ ~hultman/, 2006), together with the complex of G-symmetric phylogenetic trees T G n . Hultman shows that the complexes T G n and ̃ (Qn(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 (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact T G n is subdivided by the order complex of Qn(G). We introduce the complex of Dowling trees Tn(G) and prove that it is subdivided by the order complex of Qn(G). Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, Tn(G) is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that Tn(G) is obtained from T G n by successive coning over certain subcomplexes. It is well known that Qn(G) is shellable, and of the same dimension as T G n . We explicitly and independently calculate how many homology spheres are added in passing from T G n to Tn(G). Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that Tn(G) is intimely related to the representation theory of the top homology of Qn(G).