A note on parabolic subgroups and the Steenrod algebra action

Let p be a prime, Fp the field with p elements, V the n-dimensional vector space over Fp and Fp[V ] the symmetric algebra on the vector space V dual to V . The general linear group GLn and the mod-p Steenrod algebra A act on Fp[V ]. Let I be a partition of n and let P(I) be the parabolic subgroup of GLn associated to I. In [N. J. Kuhn and S. A. Mitchell, Proc. Amer. Math. Soc. 96 (1986), no. 1, 1–6; MR0813797 (87h:20084)] and [T. J. Hewett, Duke Math. J. 82 (1996), no. 1, 91–102; MR1387223 (97c:13002)] the invariants in Fp[V ] of the parabolic subgroups P(I) are computed, but the resulting generating set of Fp[V ]P(I) is not closed under the action of the Steenrod algebra. In the paper under review the author provides a new system of generators for Fp[V ]P(I) which turns out to be closed under the A-action. He also derives relations between the new generators and the old ones. The results obtained are then applied to make explicit, in some cases, the A-action on the new generators of Fp[V ]P(I).