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

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 a connected unstable left $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ 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 for $d= 5$ is not yet completely solved, and unknow in general.In this work, we give an explicit solution to the hit problem of five variables in the generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5,\ s =8$ and $t$ an arbitrary non-negative integer. An application of this study to the cases $t = 0$ and $t = 1$ shows that the Singer algebraic transfer of rank $5$ is an isomorphism in the bidegrees $(5, 5+(13.2^{0}-5))$ and $(5, 5+(13.2^{1}-5)).$ Moreover, the result when $t\geq 2$ was also discussed. Here, the Singer transfer of rank $d$ is a $\mathbb Z/2$-algebra homomorphism from $GL_d$-coinvariants of certain subspaces of $Q\mathscr P_d$ to the cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A_2}^{d, d+*}(\mathbb Z/2, \mathbb Z/2).$ It is one of the useful tools for studying these mysterious Ext groups.

The hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ and its application

“…The above computations confirm the dimension of (QP 5 ) ns , which is noted in [11] by using the MAGMA computer algebra system. Based upon Theorems 1.2 and 2.12, we have the following, which is our second main result.…”

“…It has been surveyed by many authors. (See Kameko [2], Mothebe-Uys [4], the present author [8,9,10,11], Singer [13], Sum [14,15], Wood [17], and others. )…”

“…The structure of QP t was explicitly described by Peterson [5] for t = 1, 2, by Kameko [2] for t = 3, and by Sum [14] for t = 4. When t = 5 and in some generic degrees of the form (1), it has been surveyed by us [6,7,8,9,10,11,15], and Tín [16]. The problem is not yet determined in general, even with the help of the computer.…”

The "hit" problem of five variables in the generic degree and its application

“…However, when s 5, it is an open problem. Recently, some authors have been studied the conjecture for s = 4, 5 (see Bruner-Hà-Hưng [4], Hưng [11], Chơn-Hà [7,8], Hà [9], Nam [22], the present author [26], [28]- [35], Sum [45,47,48,49] and others). In the present work, by using techniques of the hit problem of five variables, we investigate Conjecture 1.2 in bidegree (5, d + 5), where d is determined as in Theorem 1.1.…”

“…We explicitly compute p (i;I) (S) in terms u j , 1 j 23. By a direct computation using Lemma 2.2.9,Theorem 3.1.3, and from the relations p (i;j) (S) ≡ ω (5,2) 0 with either i = 1, j = 2, 3 or i = 2, j = 3, 4, one gets Here J = {1, 2, 3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,21,22,23,24,25,26,28,29,31,38,39,40,41,42,45,46,48,50,51,57 [46]). By a simple computation, we get…”