Reiner and Webb (preprint, 2002) compute the $S_n$-module structure for the complex of injective words. This paper refines their formula by providing a Hodge type decomposition. Along the way, this paper proves that the simplicial boundary map interacts in a nice fashion with the Eulerian idempotents.

The Laplacian acting on the top chain group in the complex of injective words is also shown to equal the signed random to random shuffle operator. Uyemura-Reyes, 2002, conjectured that the (unsigned) random to random shuffle operator has integral spectrum. We prove that this conjecture would imply that the Laplacian on (each chain group in) the complex of injective words has integral spectrum.