Nested set complexes of Dowling lattices and complexes of Dowling trees

• Authors: Emanuele Delucchi
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2007

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 $\mathcal {Q}_{n}(G)$ (Proc. Internat. Sympos., 1971, pp. 101---115) and of its subposet of the G-symmetric partitions $\mathcal {Q}_{n}^{0}(G)$ which was recently introduced by Hultman ( http://www.math.kth.se/~hultman/ , 2006), together with the complex of G-symmetric phylogenetic trees $\mathcal {T}_{n}^{G}$ . Hultman shows that the complexes $\mathcal {T}_{n}^{G}$ and $\widetilde {\Delta }(\mathcal {Q}_{n}^{0}(G))$ 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 $\mathcal {T}_{n}^{G}$ is subdivided by the order complex of $\mathcal {Q}_{n}^{0}(G)$ . We introduce the complex of Dowling trees $\mathcal {T}_{n}(G)$ and prove that it is subdivided by the order complex of $\mathcal {Q}_{n}(G)$ . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437---468, 2005) shows that, as a simplicial complex, $\mathcal {T}_{n}(G)$ is in fact isomorphic to the Bergman complex of the associated Dowling geometry.Topologically, we prove that $\mathcal {T}_{n}(G)$ is obtained from $\mathcal {T}_{n}^{G}$ by successive coning over certain subcomplexes. It is well known that $\mathcal {Q}_{n}(G)$ is shellable, and of the same dimension as $\mathcal {T}_{n}^{G}$ . We explicitly and independently calculate how many homology spheres are added in passing from $\mathcal {T}_{n}^{G}$ to $\mathcal {T}_{n}(G)$ . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301---336, 2000) shows that $\mathcal {T}_{n}(G)$ is intimely related to the representation theory of the top homology of $\mathcal {Q}_{n}(G)$ .