Nested set complexes of Dowling lattices and complexes of Dowling trees

Delucchi, Emanuele

Journal of Algebraic Combinatorics, Tome 26 (2007) no. 4, pp. 477-494.

Résumé

Summary: 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 _{ n}( G) \mathcal $Q_n$(G)$ (Proc. Internat. Sympos., 1971, pp. 101-115) and of its subposet of the $G$-symmetric partitions $Q _{ n} ^{0}( G) \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 $T _{ n} ^{ G} \mathcal $T_n^G . Hultman shows that the complexes $T _{ n} ^{ G} \mathcal $T_n^G and [$( D)$tilde]$( Q _{ n} ^{0}( G)) \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 $T _{ n} ^{ G} \mathcal $T_n^G is subdivided by the order complex of $Q _{ n} ^{0}( G) \mathcal $Q_n^0$(G)$ . We introduce the complex of Dowling trees $T _{ n}( G) \mathcal $T_n$(G)$ and prove that it is subdivided by the order complex of $Q _{ n}( G) \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, $T _{ n}( G) \mathcal $T_n$(G)$ is in fact isomorphic to the Bergman complex of the associated Dowling geometry.

Détail

Keywords: keywords posets, lattices, combinatorial blowups, building sets, nested sets, dowling lattices, complexes of trees, phylogenetic trees