The mod 2 Margolis homology of the invariants under the Sylow subgroup of the general linear group

We completely compute the mod 2 Margolis homology of the Mùi algebra Mₙ=F₂[x₁,...,xₙ]^GLn,2, i.e. the homology of Mₙ with the differential to be the Milnor operation Qⱼ, for every n and j, where GLn,2 denotes the Sylow 2-subgroup, which is also the Borel subgroup of GL(n,F₂). The motivation for this problem is that the Margolis homology of the Mùi algebra plays a key role in study of the Morava K-theory K(j)*(Sm,2) of the Sylow 2-subgroup of the symmetric group Sₘ on m letters. We prove that for $n=1$ or 2, the j-th Margolis homology of the n-th Mùi algebra is represented by elements in Mₙ² only. For n≥3, this phenomenon is no longer true, because of the occurrence of some “critical elements” h_m,s₁,...,sₖ which we create, in the homology for $0<···