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

Anders Björner ; Michelle L. Wachs

The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2

Résumé

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 by M. L. Wachs, thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let ${\cal A}$ be a central and essential hyperplane arrangement in ${\Bbb{R}}^d$. Let $R_1,\dots,R_k$ be the bounded regions of a generic hyperplane section of ${\cal A}$. We show that there are induced polytopal cycles $\rho_{R_i}$ in the homology of the proper part $\overline{L}_{\cal A}$ of the intersection lattice such that $\{\rho_{R_i}\}_{i=1,\dots,k}$ is a basis for $\widetilde{H}_{d-2} (\overline{L}_{\cal A})$. 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.

Détail

DOI : 10.37236/1860

Classification : 52C35, 52C40
Mots-clés : hyperplane arrangement, Coxeter arrangements, interpolating arrangements