## Colored partitions and a generalization of the braid arrangement.(English)Zbl 0883.52010

Electron. J. Comb. 4, No. 1, Research paper R4, 12 p. (1997); printed version J. Comb. 4, No. 1, 27-38 (1997).
Summary: We study the topology and combinatorics of an arrangement of hyperplanes in $$\mathbb{C}^n$$ that generalizes the classical braid arrangement. The arrangement plays an important role in the work of V. V. Schechtman and A. N. Varchenko on Lie algebra homology, where it appears in a generic fiber of a projection of the braid arrangement. The study of the intersection lattice of the arrangement leads to the definition of lattices of colored partitions. A detailed combinatorial analysis then provides algebro-geometric and topological properties of the complement of the arrangement. Using results on the character of $$S_n$$ on the cohomology of these arrangements we are able to deduce the rational cohomology of certain spaces of polynomials in the complement of the standard discriminant that have no root in the first $$s$$ integers.

### MSC:

 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 05E25 Group actions on posets, etc. (MSC2000) 17B55 Homological methods in Lie (super)algebras

### Keywords:

arrangement of hyperplanes; braid arrangement
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