A Note on the Hit Problem for the Steenrod Algebra in Some Degrees

Let be the modulo-2 cohomology algebra of the direct product of k copies of infinite dimensional real projective space . Then, is isomorphic to the graded polynomial algebra of k variables, in which each is of degree 1, and let be the general linear group over the prime field which acts naturally on . Here the cohomology is taken with coefficients in the prime field of two elements. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly compute the hit problem for k = 5 and the degree n=5(2s-1)+24.2s with s an arbitrary non-negative integer. Moreover, we get the dimensional results for polynomial algebra in some generic degrees in the case k=6. Note that the main results of this paper have been published online on ArXiv [ArXiv: 2103.04393, Preprint 9 pages, March 7, 2021].