The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field with two elements. The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. Equivalently, we want to find a vector space basis for $\mathbb F_2 \otimes_{\mathcal A} P_k$ in each degree. In this paper, we study a generating set for the kernel of Kameko's squaring operation $\widetilde{Sq}^0_*: \mathbb F_2 \otimes_{\mathcal A} P_k \longrightarrow \mathbb F_2 \otimes_{\mathcal A} P_k$ in a so-called generic degree. By using this result, we explicitly compute the hit problem for $k=5$ in the respective generic degree.