## WHITNEY HOMOLOGY OF SEMIPURE SHELLABLE POSETS

**
Michelle Wachs**
**Abstract:**
We generalize results of Calderbank, Hanlon
and Robinson on the representation of the symmetric group
on the homology of posets of partitions with restricted block size.
Calderbank, Hanlon and Robinson consider the cases of block
sizes that are congruent to $0
\mod d$ and $1 \mod d$ for fixed $d$. We derive a general formula
for the representation of the symmetric group on the homology of
posets of partitions whose block sizes are congruent to
$k
\mod d$ for any $k$ and $d$. This formula reduces to the
Calderbank-Hanlon-Robinson formulas when $k=0,1$ and to formulas of
Sundaram for the virtual representation on the alternating sum of
homology. Our results apply to restricted block size partition posets
even more general than the
$k
\mod d$ partition posets. These posets include the lattice of
partitions whose block sizes are bounded from below by some fixed $k$.
Our main tools involve the new theory of nonpure shellability
developed by Bj\"orner and Wachs and a generalization of a technique
of Sundaram which uses Whitney homology to compute homology
representations of
Cohen-Macaulay posets. An application to subspace arrangements
is also discussed.

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