The action of the Steenrod squares on the modular invariants of linear groups

Abstract

We compute the action of the Steenrod squares on the Dickson invariants of the group G L n = G L ( n , Z / 2 ) G{L_n} = GL(n,{\mathbf {Z}}/2) and the Mùi invariants of the subgroup T n {T_n} consisting of all upper triangular matrices with 1 on the main diagonal. Our method is very elementary. Roughly speaking, we read off the above action from the expansion of the Mùi invariants in terms of Dickson and Mùi invariants of fewer variables.