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 \(\mathcal {Q}_{n}(G)\) (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 \(\mathcal {Q}_{n}(G)\) .
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References
Ardila, F. (November 2006). Personal communication.
Ardila, F., & Klivans, C. (2006). The Bergman complex of a matroid and phylogenetic trees. Journal of Combinatorial Theory Ser. B, 96(1), 38–49.
Billera, L. J., Holmes, S. P., & Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4), 733–767.
Björner, A. (1980). Shellable and Cohen–Macaulay partially ordered sets. Transactions of American Mathematics Society, 260, 159–183.
Boardman, J. M. (1971). Homotopy structures and the language of trees. In Proceedings of symposia in pure mathematics (Vol. 22, pp. 37–58). Providence: Am. Math. Soc.
Čukić, S., & Delucchi, E. (2006). Shellable simplicial spheres via combinatorial blowups. ArXiv math.CO/0602101. To appear in Proceedings of the American Mathematical Society.
Delucchi, E. (2005). Subdivision of complexes of k-trees. ArXiv math.CO/0509378.
Dowling, T. A. (1971). A q-analog of the partition lattice. A survey of combinatorial theory. In Proceedings of international symposium (pp. 101–115). Colorado State University.
Dowling, T. A. (1973). A class of geometric lattices based on finite groups. Journal of Combinatorial Theory Ser. B, 14, 61–86 (Erratum: Journal of Combinatorial Theory Ser. B 15, 211 (1973)).
Ehrenborg, R., & Readdy, M. A. (2000). The Dowling transform of subspace arrangements. Journal of Combinatorial Theory Ser. A, 91(1–2), 322–333.
Ehrenborg, R., & Readdy, M. A. (1999). On flag vectors, the Dowling lattice, and braid arrangements. Discrete & Computational Geometry, 21(3), 389–403.
Feichtner, E. M. (2006). Complexes of trees and nested set complexes. Pacific Journal of Mathematics, 227, 271–286. ArXiv math.CO/0409235.
Feichtner, E. M., & Kozlov, D. N. (2003). Abelianizing the real permutation action via blowups. International Mathematics Research Notices, 32, 1755–1784.
Feichtner, E. M., & Kozlov, D. N. (2004). Incidence combinatorics of resolutions. Selecta Mathematica (N.S.), 10(1), 37–60.
Feichtner, E. M., & Müller, I. (2005). On the topology of nested set complexes. Proceedings of American Mathematical Society, 133(4), 999–1006.
Feichtner, E. M., & Sturmfels, B. (2005). Matroid polytopes, nested sets, and Bergman fans. Portugaliae Mathematica (N.S.), 62, 437–468. ArXiv math.CO/0411260.
Feichtner, E. M., & Yuzvinsky, S. (2004). Chow rings of toric varieties defined by atomic lattices. Inventiones Mathematicae, 155(3), 515–536.
Gottlieb, E., & Wachs, M. (2000). Cohomology of Dowling lattices and Lie (super)algebras. Advances in Applied Mathematics, 24(4), 301–336.
Hanlon, P. (1996). Otter’s method and the homology of homeomorphically irreducible k-trees. Journal of Combinatorial Theory Ser. A, 74(2), 301–320.
Hanlon, P. (1991). The generalized Dowling lattices. Transactions of American Mathematics Society, 325(1), 1–37.
Hanlon, P., & Wachs, M. (1995). On Lie k-algebras. Advances in Mathematics, 113(2), 206–236.
Hultman, A. (2006). The topology of spaces of phylogenetic trees with symmetry. Preprint available at http://www.math.kth.se/~hultman/. To appear in Discrete Mathematics.
Robinson, A., & Whitehouse, S. (1996). The tree representation of Σ n+1. Journal of Pure Applied Algebra, 111(1–3), 245–253.
Stanley, R. P. (1972). Supersolvable lattices. Algebra Universalis, 2, 197–217.
Trappmann, H., & Ziegler, G.M. (1998). Shellability of complexes of trees. Journal of Combinatorial Theory Ser. A, 82, 168–178.
Vogtmann, K. (1990). Local structure of some Out(F n )-complexes. Proceedings of the Edinburgh Mathematical Society (2), 33(3), 367–379.
Welsh, D. J. A. (1976) Matroid theory. L.M.S. monographs (Vol. 8). London: Academic
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Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.
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Delucchi, E. Nested set complexes of Dowling lattices and complexes of Dowling trees. J Algebr Comb 26, 477–494 (2007). https://doi.org/10.1007/s10801-007-0067-2
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DOI: https://doi.org/10.1007/s10801-007-0067-2