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Homology representations arising from the half cube, II
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes
of the -dimensional half cube by removing the interiors of all half cube
shaped faces of dimension at least , and we proved that the homology of such
a subcomplex is concentrated in degree . This homology group supports a
natural action of the Coxeter group of type . In this paper, we
explicitly determine the characters (over ) of these homology
representations, which turn out to be multiplicity free. Regarded as
representations of the symmetric group by restriction, the homology
representations turn out to be direct sums of certain representations induced
from parabolic subgroups. The latter representations of \sym_n agree (over
) with the representations of \sym_n on the -nd homology of
the complement of the -equal real hyperplane arrangement.Comment: 19 pages AMSTeX. One figure. The Conjecture in the previous version
is now a Theorem. This research was supported by NSF grant DMS-090576
On rank functions for heaps
Motivated by work of Stembridge, we study rank functions for Viennot's heaps
of pieces. We produce a simple and sufficient criterion for a heap to be a
ranked poset and apply the results to the heaps arising from fully commutative
words in Coxeter groups.Comment: 18 pages AMSTeX, 3 figure
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