### 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) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 11 |

Issue number | 2 R |

State | Published - Jun 3 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

**Geometrically constructed bases for homology of partition lattices of types A, B and D.** / Björner, Anders; Galloway, Michelle L.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 11, no. 2 R.

}

TY - JOUR

T1 - Geometrically constructed bases for homology of partition lattices of types A, B and D

AU - Björner, Anders

AU - Galloway, Michelle L

PY - 2004/6/3

Y1 - 2004/6/3

N2 - 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 ℝd. Let R 1, . . . ,Rk be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρRi in the homology of the proper part L̄A of the intersection lattice such that {ρRi}i=1,...,k is a basis for H̃d-2(L̄A). 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.

AB - 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 ℝd. Let R 1, . . . ,Rk be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles ρRi in the homology of the proper part L̄A of the intersection lattice such that {ρRi}i=1,...,k is a basis for H̃d-2(L̄A). 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.

UR - http://www.scopus.com/inward/record.url?scp=3042675809&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042675809&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042675809

VL - 11

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2 R

ER -