A Hodge Decomposition for the Complex of Injective Words

    Patricia Hersh, Univ. of Michigan Ann Arbor



I will discuss two recent results about the complex
of injective words, both of which are joint work with
Phil Hanlon.  The first is a Hodge-type decomposition
for the S_n-module structure for the (top) homology,
refining a recent formula of Reiner and Webb.  A key
ingredient was to show that the Eulerian idempotents
interact in a nice fashion with the simplicial boundary
map on the complex of injective words.  The second
result deals with a recent conjecture of Uyemura-Reyes,
namely that the 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 also has integral
spectrum.