On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra

Abstract

Denote by $P_k$ the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ over the prime field of two elements, ${\mathbb{F}}_2$, with the degree of each $x_i$ being 1. We study the Peterson hit problem of determining a minimal set of generators for $P_k$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_k$ in the case $k=5$ and the degree $4(2^d-1)$ with $d$ an arbitrary positive integer.

21 pages, This is a revision of a preprint of Quy Nhon University, Viet Nam, 2013. A shorter version of this paper was accepted for publication in Acta Mathematica Vietnamica. arXiv admin note: text overlap with arXiv:1412.3309

References in corpus (3)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• On the Peterson hit problem
• Young tableaux and the Steenrod algebra

Cited by in corpus (8)

• The "hit" problem of five variables in the generic degree and its application
• On a construction for the generators of the polynomial algebra as a module over the Steenrod algebra
• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• A note on the hit problem for the Steenrod algebra and its applications
• The squaring operartion and the Singer algebraic transfer
• A note on the non-trivial elements in the cohomology groups of the Steenrod algebra
• The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees
• On modules over the mod 2 Steenrod algebra and hit problems