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.