A note on the generators of the polynomial algebra of six variables and application

• Published in: Carpathian Journal of Mathematics 2023-01, Vol.39 (2), p.529-539
• Main Author: Tin, Nguyen Khac
• Format: Article
• Language: English
• ISSN: 1584-2851; 1843-4401
• DOI: 10.37193/CJM.2023.02.13

Let P n := H*((RP ∞) n ) ≅ Z2[x1, x2,..., xn] be the graded polynomial algebra over K, where K denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra Pn, viewed as a graded left module over the mod-2 Steenrod algebra, A. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this paper, we study the hit problem for the case n = 6 in degree dk = 6(2 k − 1) + 9.2 k , with k an arbitrary non-negative integer. By considering K as a trivial A-module, then the hit problem is equivalent to the problem of finding a basis of K-graded vector space K⊗AP n . The main goal of the current paper is to explicitly determine an admissible monomial basis of the K-graded vector space K⊗AP6 in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree dk = 6(2 k − 1) + 9.2 k is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, Tor n , n + d A ( K , K ) , to the subspace of K ⊗A Pn consisting of all the GLn -invariant classes of degree d.