On certain homotopy actions of general linear groups on iterated products

The n-fold product X-n of an arbitrary space usually supports only the obvious permutation action of the symmetric group Sigma(n). However, if X is a p-complete, homotopy associative, homotopy commutative H-space one can define a homotopy action of CLn(Z(p)) on X-n. In various cases, e.g. if multiplication by p(r) is null homotopic then we get a homotopy action of GLn(Z/(T)(p)) for some r. After one suspension this allows one to split X-n using idempotents of F(p)GL(n)(Z/p) which can be lifted to F(P)GL(n)(Z/(r)(p)). In fact all of this is possible if X is an H-space whose homology algebra H-* (X; Z/(p)) is commutative and nilpotent. For n = 2 we make some explicit calculations of splittings of Sigma(SO (4) xSO(4)), Sigma(Omega(2)S(3) XOmega(2)S(3)) and Sigma(G(2) x G(2)).