Nested set complexes of Dowling lattices and complexes of Dowling trees
Emanuele Delucchi
Università di Pisa Dipartimento di Matematica Largo Bruno Pontecorvo 5 56127 Pisa Italy
DOI: 10.1007/s10801-007-0067-2
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 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 {Δ}(\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.
Pages: 477–494
Keywords: keywords posets; lattices; combinatorial blowups; building sets; nested sets; dowling lattices; complexes of trees; phylogenetic trees
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References
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11. Ehrenborg, R., & Readdy, M. A. (1999). On flag vectors, the Dowling lattice, and braid arrangements. Discrete & Computational Geometry, 21(3), 389-403.
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13. Feichtner, E. M., & Kozlov, D. N. (2003). Abelianizing the real permutation action via blowups. International Mathematics Research Notices, 32, 1755-1784.
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15. Feichtner, E. M., & Müller, I. (2005). On the topology of nested set complexes. Proceedings of American Mathematical Society, 133(4), 999-1006.
16. Feichtner, E. M., & Sturmfels, B. (2005). Matroid polytopes, nested sets, and Bergman fans. Portugaliae Mathematica (N.S.), 62, 437-468. ArXiv math.CO/0411260.
17. Feichtner, E. M., & Yuzvinsky, S. (2004). Chow rings of toric varieties defined by atomic lattices. Inventiones Mathematicae, 155(3), 515-536.
18. Gottlieb, E., & Wachs, M. (2000). Cohomology of Dowling lattices and Lie (super)algebras. Advances in Applied Mathematics, 24(4), 301-336.
19. Hanlon, P. (1996). Otter's method and the homology of homeomorphically irreducible k-trees. Journal of Combinatorial Theory Ser. A, 74(2), 301-320.
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21. Hanlon, P., & Wachs, M. (1995). On Lie k-algebras. Advances in Mathematics, 113(2), 206-236.
22. 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.
23. Robinson, A., & Whitehouse, S. (1996). The tree representation of n+1. Journal of Pure Applied Algebra, 111(1-3), 245-253.
24. Stanley, R. P. (1972). Supersolvable lattices. Algebra Universalis, 2, 197-217.
25. Trappmann, H., & Ziegler, G.M. (1998). Shellability of complexes of trees. Journal of Combinatorial Theory Ser. A, 82, 168-178.
26. Vogtmann, K. (1990). Local structure of some Out(Fn)-complexes. Proceedings of the Edinburgh Mathematical Society (2), 33(3), 367-379.
2. Ardila, F., & Klivans, C. (2006). The Bergman complex of a matroid and phylogenetic trees. Journal of Combinatorial Theory Ser. B, 96(1), 38-49.
3. Billera, L. J., Holmes, S. P., & Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4), 733-767.
4. Björner, A. (1980). Shellable and Cohen-Macaulay partially ordered sets. Transactions of American Mathematics Society, 260, 159-183.
5. 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.
6. \check Cukić, S., & Delucchi, E. (2006). Shellable simplicial spheres via combinatorial blowups. ArXiv math.CO/0602101. To appear in Proceedings of the American Mathematical Society.
7. Delucchi, E. (2005). Subdivision of complexes of k-trees. ArXiv math.CO/0509378.
8. 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.
9. 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)).
10. Ehrenborg, R., & Readdy, M. A. (2000). The Dowling transform of subspace arrangements. Journal of Combinatorial Theory Ser. A, 91(1-2), 322-333.
11. Ehrenborg, R., & Readdy, M. A. (1999). On flag vectors, the Dowling lattice, and braid arrangements. Discrete & Computational Geometry, 21(3), 389-403.
12. Feichtner, E. M. (2006). Complexes of trees and nested set complexes. Pacific Journal of Mathematics, 227, 271-286. ArXiv math.CO/0409235.
13. Feichtner, E. M., & Kozlov, D. N. (2003). Abelianizing the real permutation action via blowups. International Mathematics Research Notices, 32, 1755-1784.
14. Feichtner, E. M., & Kozlov, D. N. (2004). Incidence combinatorics of resolutions. Selecta Mathematica (N.S.), 10(1), 37-60. J Algebr Comb (2007) 26: 477-494
15. Feichtner, E. M., & Müller, I. (2005). On the topology of nested set complexes. Proceedings of American Mathematical Society, 133(4), 999-1006.
16. Feichtner, E. M., & Sturmfels, B. (2005). Matroid polytopes, nested sets, and Bergman fans. Portugaliae Mathematica (N.S.), 62, 437-468. ArXiv math.CO/0411260.
17. Feichtner, E. M., & Yuzvinsky, S. (2004). Chow rings of toric varieties defined by atomic lattices. Inventiones Mathematicae, 155(3), 515-536.
18. Gottlieb, E., & Wachs, M. (2000). Cohomology of Dowling lattices and Lie (super)algebras. Advances in Applied Mathematics, 24(4), 301-336.
19. Hanlon, P. (1996). Otter's method and the homology of homeomorphically irreducible k-trees. Journal of Combinatorial Theory Ser. A, 74(2), 301-320.
20. Hanlon, P. (1991). The generalized Dowling lattices. Transactions of American Mathematics Society, 325(1), 1-37.
21. Hanlon, P., & Wachs, M. (1995). On Lie k-algebras. Advances in Mathematics, 113(2), 206-236.
22. 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.
23. Robinson, A., & Whitehouse, S. (1996). The tree representation of n+1. Journal of Pure Applied Algebra, 111(1-3), 245-253.
24. Stanley, R. P. (1972). Supersolvable lattices. Algebra Universalis, 2, 197-217.
25. Trappmann, H., & Ziegler, G.M. (1998). Shellability of complexes of trees. Journal of Combinatorial Theory Ser. A, 82, 168-178.
26. Vogtmann, K. (1990). Local structure of some Out(Fn)-complexes. Proceedings of the Edinburgh Mathematical Society (2), 33(3), 367-379.
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