Permutation Modules associated to the Hyperoctahedron and Group Actions

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.