On Peterson's open problem and representations of the general linear groups

J. Korean Math. Soc. Published online February 8, 2021

Đặng Võ Phúc
University of Khanh Hoa, 01 Nguyen Chanh, Nha Trang, Khanh Hoa, Vietnam

Abstract : Fix $\mathbb Z/2$ is the prime field of two elements and write $\mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, \mathbb Z/2)$ the general linear group of rank $d$ over $\mathbb Z/2$ and by $\mathscr P_d$ the polynomial algebra $\mathbb Z/2[x_1, x_2, \ldots, x_d]$ as an unstable left $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson hit problem of finding a minimal set of generators for $\mathcal A_2$-module $\mathscr P_d.$ Equivalently, we need to determine a basis for the $\mathbb Z/2$-vector space
$$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d$$
in each degree $n\geq 1.$ Note that this space is a representation of $GL_d$ over $\mathbb Z/2.$ The problem is still open for $d\geq 5.$

In this paper, we compute the hit problem in the $5$-variable case and in generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary positive integer. This result is used to prove that the fifth Singer algebraic transfer is an isomorphism in the respective degree for $t = 1.$ This transfer is a homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, *}(\mathbb Z/2, \mathbb Z/2)$.

Keywords : Steenrod algebra; Peterson hit problem; Singer algebraic transfer