Bases for certain cohomology representations of the symmetric group

Details Bases for certain cohomology representations of the symmetric group Bases for certain cohomology representations of the symmetric group

2005-08-09

arxiv.org/abs/math/0508162v3

J. Algebraic Combin. 24 (2006), no. 4, pp. 361-390.

Mathematics

Representation Theory

30 pages; in third version, more references are added, making further connections with general theory of cohomology of local s

Scientific paper

10.1007/s10801-006-0018-3

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.

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