Related papers: On Singer's conjecture for the fifth algebraic tra…

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field with two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a…

Let $A$ be the Steenrod algebra over the finite field $k := \mathbb Z_2$ and $G(q)$ be the general linear group of rank $q$ over $k.$ A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of…

Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$…

In recent decades, the structure of the mod-2 cohomology of the Steenrod ring $\mathscr A$ has become a major subject of study in the field of Algebraic Topology. One of the earliest attempts to study this cohomology through the use of…

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…

Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2.$ Denote by $GL(q)$ the general linear group of rank $q$ over $\mathbb F_2.$ The algebraic transfer, introduced by W. Singer [Math. Z. 202 (1989),…

The Singer algebraic transfer is a fundamental homomorphism in algebraic topology, providing a bridge between the homology of classifying spaces and the cohomology of the Steenrod algebra $\mathcal{A}$, which forms the $E_2$-term of the…

We study the problem of determining a minimal set of generators for the polynomial algebra $\mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. In this paper, we give an explicit answer in terms of the…

Let V be a mod 2 vector space of rank k. W. Singer defined a transfer homomorphism from the GL(k,2) coinvariants of the primitives in the homology of BV to the cohomology of the Steenrod algebra, as an algebraic version of the geometric…

Writing $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an…

Let $R_s M$ denote the Singer construction on an unstable module $M$ over the Steenrod algebra $A$ at the prime two; $R_s M$ is canonically a subobject of $P_s\otimes M$, where $P_s$ is the polynomial algebra on s generators of degree one.…

The Peterson hit problem in algebraic topology is to explicitly determine the dimension of the quotient space $Q\mathcal P_k = \mathbb F_2\otimes_{\mathcal A}\mathcal P_k$ in positive degrees, where $\mathcal{P}_k$ denotes the polynomial…

Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $\Lambda,$ is one of the tools to describe…

The Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of `Quillen stratification' theorem for the Singer transfer and its consequences.

One knows that, the connected graded ring $P^{\otimes h}= \mathbb Z/2[t_1, \ldots, t_h]= \{P_n^{\otimes h}\}_{n\geq 0},$ which is graded by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the…

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…

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by…

G. Walker and R. Wood proved that in degree $2^n-1-n$, the space of indecomposable elements of $\Bbb F_2[x_1,\ldots,x_n]$, considered as a module over the mod 2 Steenrod algebra, is isomorphic to the Steinberg representation of $GL_n(\Bbb…

We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the…

The aim of this note is to use geometric methods to study the hit problem of Peterson for $H_*\mathbb{R} P^{\times k}$ as well as the symmetric hit problem of Janfada and Wood for $H_*BO(k)$. We continue by exploring the applications of the…