Group representations on the homology of products of posets (Q1906130)

Group actions on homologies of posets have been studied from various viewpoints in recent years, see, for example, \textit{R. P. Stanley} [J. Comb. Theory, Ser. A 32, 132-161 (1982; Zbl 0496.06001)] and \textit{G. I. Lehrer} and \textit{L. Solomon} [J. Algebra 104, 410-424 (1986; Zbl 0608.20010)]. Here the following problem is considered: Assume that \(G\) is a group of automorphisms of the poset \(P\) and that the corresponding representation of \(G\) on the homology of \(P\) is known. Let the symmetric group \(S_n\) act on the \(n\)-fold power \(P^n:= P\times\cdots\times P\) by permuting the coordinates. The action of these two groups induces an action of the wreath product \(S_n[G]\) on \(P^n\). What is the corresponding homology representation and its character? The authors answer to the question and apply this to Boolean lattices and lattices of partitions.