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

Let $P_{k}=\oplus_{n\geqslant 0} (P_{k})_{n} \cong \mathbf{F}_{2}[x_{1},x_{2},\ldots ,x_{k}]$ be the graded polynomial algebra over the prime field of two elements $\mathbf{F}_{2}$, in $k$ generators $x_{1}, x_{2}, \ldots , x_{k}$, each of degree 1. Being the mod-2 cohomology of the classifying space $B(\mathbf{Z}/2)^{k}$, the algebra $P_{k}$ is a module over the mod-2 Steenrod algebra $\mathcal{A}$. In this Note, we explicitly compute the hit problem of some generic degrees $r(2^{s}-1)+2^{s}m$ in $P_{k}$, where $r=k-1=4, m \in \{8; 10; 11 \}$ and $s$ an arbitrary non-negative integer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees and in the cases $k=5$ and 6.

Let 𝓟 n := H *((ℝP ∞) n ) ≅ ℤ2[x 1, x 2, …, x n ] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 𝓟 n , 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 article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2 r − 1) + 4.2 r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗𝓐𝓟 n . The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗𝓐𝓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2 r − 1) + 4.2 r is also discussed at the end of this paper.

Let $\mathcal P_n \cong H^{*}\big(BE_n; \mathbb F_2 \big)$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This problem remains unsolvable for $n>4,$ even with the aid of computers in the case of $n=5.$ By considering $\mathbb F_2$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathbb F_2$-graded vector space $\mathbb F_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ This paper aims to explicitly determine an admissible monomial basis of the $ \mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ in the generic degree $n(2^{r}-1)+2\cdot 2^{r},$ where $r$ is an arbitrary non-negative integer, and in the case of $n=6.$ As an application of these results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degrees $(n-1)\cdot(2^{n+u-1}-1)+\ell\cdot2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$ Moreover, for any integer $r>1,$ the behavior of the sixth Singer algebraic transfer in degree $6(2^{r}-1)+2\cdot2^r$ is also discussed at the end of this paper. Here, the Singer algebraic transfer is a homomorphism from the homology of the Steenrod algebra to the subspace of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n(\mathbb F_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+*}(\mathbb F_2,\mathbb F_2).$

On the Peterson hit problem

We study the hit problem, set up by F. Peterson of finding a minimal set of generators for the polynomial algebra ( ) 2 1 2 , ,..., kk P F x x x = as a module over the mod-2 Steenrod algebra, A. By assigning degree 1 to each ,1 i x i k ≤≤ , k P is regarded as a graded algebra over the ground field 2 F. The mod 2 cohomology ring of the k-fold Cartesian product of infinite dimensional real projective spaces is isomorphic to k P as a graded algebra. Through this isomorphism, we may regard k P as an A-module where A stands for the mod 2 Steenrod algebra. In this paper, we explicitly determine the hit problem for the case of k=5 in degree 8 in terms of the admissible monomials.

The hit problem for the polynomial algebra in some weight vectors

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

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables with the degree of each [Formula: see text] being [Formula: see text] regarded as a module over the mod-[Formula: see text] Steenrod algebra [Formula: see text] and let [Formula: see text] be the general linear group over the prime field [Formula: see text] which acts naturally on [Formula: see text]. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra [Formula: see text] as a module over the mod-2 Steenrod algebra, [Formula: see text]. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-[Formula: see text] Steenrod algebra, [Formula: see text] to the subspace of [Formula: see text] consisting of all the [Formula: see text]-invariant classes of degree [Formula: see text] In this paper, we explicitly compute the hit problem for [Formula: see text] and the degree [Formula: see text] with [Formula: see text] an arbitrary positive integer. Using this result, we show that Singer’s conjecture for the algebraic transfer is true in the case [Formula: see text] and the above degree.

On the generators of the polynomial algebra as a module over the Steenrod algebra

The negative answer to Kameko's conjecture on the hit problem

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 $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ It is equivalent to determining a $\mathbb Z/2$-basis for the space of "cohits" $Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d.$ This $Q\mathscr P_d$ is also a representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknown 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 mysterious Ext groups and the Kervaire invariant one problem.

The hit problem, set up by F. Peterson, finds a minimal set of generators for the polynomial algebra , as a module over the mod-2 Steenrod algebra. We study the extended hit problem for the cohomology of the classifying space over field , with at degrees d

Let $P_n$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_n]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field of two elements. The Peterson hit problem is to find a minimal generating set for $P_n$ regarded as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. Equivalently, we want to find a vector space basis for $\mathbb F_2 \otimes_{\mathcal A} P_n$ in each degree $d$. Such a basis may be represented by a list of monomials of degree $d$. In this paper, we present a construction for the $\mathcal A$-generators of $P_n$ and prove some properties of it. We also explicitly determine a basis of $\mathbb F_2 \otimes_{\mathcal A} P_n$ for $n = 5$ and the degree $d = 15.2^s - 5$ with $s$ an arbitrary positive integer. These results are used to verify Singer's conjecture for the fifth Singer algebraic transfer in respective degree.

The purpose of this paper is to forge a direct link between the hit problem for the action of the Steenrod algebra A on the polynomial algebra P(n)=F_2[x_1,...,x_n], over the field F_2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F_2). The cohits Q^d(n)=P^d(n)/P^d(n)\cap A^+(P(n)) form a modular representation of GL(n,F_2) and the hit problem is to analyze this module. In certain generic degrees d we show how the semistandard Young tableaux can be used to index a set of monomials which span Q^d(n). The hook formula, which calculates the number of semistandard Young tableaux, then gives an upper bound for the dimension of Q^d(n). In the particular degree d where the Steinberg module appears for the first time in P(n) the upper bound is exact and Q^d(n) can then be identified with the Steinberg module.

On the hit problem for the polynomial algebra