First Author: Victor Reiner
email: reiner@math.umn.edu
Address:
School of Mathematics,
University of Minnesota,
Minneapolis MN 55455
Second Author: Peter Webb
email: webb@math.umn.edu
Address:
School of Mathematics,
University of Minnesota,
Minneapolis MN 55455
Title:
Combinatorics of the bar resolution: the complex of words without repetition,
a derangement representation, and a spectral sequence in the cohomology
of groups
Abstract:
We study a combinatorially-defined double complex structure on the
ordered chains of any simplicial complex. Its columns turn out to be
related to the cell complex $K_n$ whose face poset is isomorphic to the
subword ordering on words without repetition from an alphabet of size
$n$. This complex is known to be shellable and we provide two
applications of this fact.
First, the action of the symmetric group on the homology of $K_n$ gives
a representation theoretic interpretation for derangement numbers and a
related symmetric function considered by D\'esarm\'enien and Wachs.
Second, the vanishing of homology below the top dimension for $K_n$ and
the links of its faces provides a crucial step in understanding one of
the two spectral sequences associated to the double complex.
We analyze also the other spectral sequence arising from the double
complex in the case of the bar resolution for a group. This spectral
sequence converges to the cohomology of the group and provides a method
for computing group cohomology in terms of the cohomology of subgroups.
Its behavior is influenced by the complex of oriented chains of the
simplicial complex of finite subsets of the group, and we examine the Ext
class of this complex.
Journal: J. Pure Appl. Algebra 190 (2004), 291-327.