Group Representations on the Homology of Products of Posets

In this note we give a description of the representation of the wreath product Sn[G] of the symmetric group Sn and a finite group G on the homology of the product of n copies of a partially ordered set (poset for short) P on which G acts as a group of automorphisms. In the sequel all posets will be finite. We will consider three types of product constructions. The direct product P ×Q of two posets P and Q is the poset on the Cartesian product P ×Q whose order relation is defined by (x, y) ≤ (x′, y′) if and only if x ≤ x′ in P and y ≤ y′ in Q. In case a poset P contains a least element 0̂, we write P ×0 · · · ×0 P } {{ } n times for P × · · · × P } {{ }−{(0̂, . . . , 0̂)}. If, addition-