### Abstract

We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the "splitting basis" for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in ℝ^{d}. Let R _{1}, . . . ,R_{k} be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρ_{Ri} in the homology of the proper part L̄_{A} of the intersection lattice such that {ρ_{Ri}}_{i=1,...,k} is a basis for H̃_{d-2}(L̄_{A}). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.

Original language | English (US) |
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Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 2 R |

DOIs | |

State | Published - Jun 3 2004 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics