  We investigate the permutation modules associated to the set of
$k$-dimensional faces of the hyperoctahedron in dimension $n$, denoted $H^{n}.$
For any $k\leq n$ such a module can be defined over an arbitrary field $F$, it
is called a face module of $H^{n}$ over $F.$ We describe a spectral
decomposition of such face modules into submodules and show that these
submodules are irreducible under the hyperoctahedral group $B_{n}.$ The same
method can be used to describe the exact relationship between the face modules
in any two dimensions $0\leq t\leq k\leq n.$ Applications of this technique
include a rank formula for the rank of the incidence matrix of $t$-dimensional
versus $k$-dimensional faces of $H^{n}$ and a characterization of
$(t,k,\ell)$-designs on $H^{n}.$ We also prove an orbit theorem for subgroups
of the hyperoctahedral group on the set of faces of $H^{n}.$ The decomposition
method is elementary, mostly characteristic free and does not involve the
representation theory of automorphism groups. It is therefore quite general and
can be used to decompose permutation modules associated to other geometries.
