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

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

2004-01-02

arxiv.org/abs/math/0401006v1

Mathematics

Combinatorics

29 pages, 4 figures

Scientific paper

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 [Wa96], 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 \rho_{R_i} in the homology of the proper part \bar{L_A} of the intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde H_{d-2}(\bar{L_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.

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