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Homotopy of Non-Modular Partitions and the Whitehouse Module
| Content Provider | Semantic Scholar |
|---|---|
| Author | Sundaram, Sheila |
| Copyright Year | 1999 |
| Abstract | AbstractWe present a class of subposets of the partition lattice Πn with the following property: The order complex is homotopy equivalent to the order complex of Πn − 1, and the Sn-module structure of the homology coincides with a recently discovered lifting of the Sn − 1-action on the homology of Πn − 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet Pnn − 1 of the lattice of set partitions Πn, obtained by removing all elements with a unique nontrivial block. More generally, for 2 ≤ k ≤ n − 1, let Qnk denote the subposet of the partition lattice Πn obtained by removing all elements with a unique nontrivial block of size equal to k, and let Pnk = ∩i = 2kQni. We show that Pnk is Cohen-Macaulay, and that Pnk and Qnk are both homotopy equivalent to a wedge of spheres of dimension (n − 4), with Betti number $$(n - 1)!\frac{{n - k}}{k}$$ . The posets Qnk are neither shellable nor Cohen-Macaulay. We show that the Sn-module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property. |
| Starting Page | 251 |
| Ending Page | 269 |
| Page Count | 19 |
| File Format | PDF HTM / HTML |
| DOI | 10.1023/A:1018648219348 |
| Alternate Webpage(s) | http://www.maths.tcd.ie/EMIS/journals/JACO/Volume9_3/nk1341341321w435.fulltext.pdf |
| Alternate Webpage(s) | http://www.mathe2.uni-bayreuth.de/axel/papers/sundaram:homotopy_of_non_modular_partitions_and_the_whitehouse_module.mar97.ps.gz |
| Alternate Webpage(s) | http://www.mathe2.uni-bayreuth.de/axel/papers/sundaram:homotopy_of_non_modular_partitions_and_the_whitehouse_module.sept97.ps.gz |
| Alternate Webpage(s) | https://doi.org/10.1023/A%3A1018648219348 |
| Volume Number | 9 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
