The smallest subgroup whose invariants are hit by the Steenrod algebra

2007 ◽  
Vol 142 (1) ◽  
pp. 63-71
Author(s):  
NGUYỄN H. V. HƯNG ◽  
TRAN DINH LUONG
Keyword(s):  
Vector Space ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Vector Subspace ◽  
Dimensional Vector ◽  
Hopf Invariant ◽  
Algebraic Version

AbstractLet V be a k-dimensional ${\mathbb{F}_2}$-vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, ${\mathbb{F}_2}$) • 1k-n the subgroup of GL(V) consisting of all isomorphisms ϕ:V → V with ϕ(W) = W and ϕ(v) ≡ v (mod W) for every v ∈ V. We show that GL(3, ${\mathbb{F}_2}$) • 1k-3 is, in some sense, the smallest subgroup of GL(V)≅ GL(k, ${\mathbb{F}_2})$, whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, ${\mathbb{F}_2})\cong{\mathbb{F}_2}[x_{1},\ldots,x_{k}]$. The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes inQ0S0are the elements of Hopf invariant one and those of Kervaire invariant one.

scholarly journals On the dimension of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} H^{*}((\mathbb RP(\infty))^{\times t}, \mathbb Z_2)$ and some applications

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Negative Integer ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Monomial Basis ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with the degree of each $x_i$ being one. Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\mathbb Z_2\otimes_{\mathcal A_2} P_t.$ This problem, which is called the "hit" problem for Steenrod algebra, has been systematically studied for $t\leq 4.$ The present paper is devoted to the investigation of the structure of the "cohits" space $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in some certain "generic" degrees. More specifically, we explicitly determine a monomial basis of $\mathbb Z_2\otimes_{\mathcal A_2} P_5$ in degree \mbox{$n_s=5(2^{s}-1) + 42.2^{s}$} for every non-negative integer $s.$ As a result, it confirms Sum's conjecture \cite{N.S2} for a relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n_s$. Based on Kameko's map \cite{M.K} and a previous result by Sum \cite{N.S1}, we obtain a inductive formula for the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_t$ in a generic degree given. As an application, we obtain the dimension of $\mathbb Z_2\otimes_{\mathcal A_2} P_6$ in the generic degree $5(2^{s+5}-1) + n_0.2^{s+5}$ for all $s\geq 0,$ and show that the Singer's cohomological transfer \cite{W.S1} is an isomorphism in bidegree $(5, 5+n_s)$.

scholarly journals Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

2016 ◽  
Vol 8 (3) ◽  
pp. 112
Author(s):  
Mbakiso Fix Mothebe
Keyword(s):  
Vector Space ◽  
Algebraic Topology ◽  
Related Problem ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Homogeneous Polynomials ◽  
Sigma 1

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules.  A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$.  Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. ${\Q}(n)$ has been explicitly calculated for $n=1,2,3,4$ but problems remain for $n \geq 5.$ In this note we show that if  $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$  and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ are an admissible  monomials, (that is,  $u$ and $v$ meet a criterion to be in a certain basis for ${\Q}(k)$ and ${\Q}(r)$ respectively), then for each permutation $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible.  As an application we consider a few cases when $n=5.$

scholarly journals Dimension Formulae for the Polynomial Algebra as a Module over the Steenrod Algebra in Degrees Less than or Equal to $12$

2016 ◽  
Vol 8 (5) ◽  
pp. 92
Author(s):  
Mbakiso Fix Mothebe ◽  
Professor Kaelo ◽  
Orebonye Ramatebele
Keyword(s):  
Vector Space ◽  
Algebraic Topology ◽  
Related Problem ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Homogeneous Polynomials ◽  
Explicit Formulae

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules.  A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. In this paper we give explicit formulae for the dimension of ${\Q}(n)$ in degrees less than or equal  to $12.$

scholarly journals On the ranks of some (0, 1)-matrices with constant row sums

1981 ◽  
Vol 31 (2) ◽  
pp. 193-201 ◽  
Author(s):  
A. M. Odlyzko
Keyword(s):  
Vector Space ◽  
Vector Subspace ◽  
Dimensional Vector ◽  
Image Position

AbstractLet g(n, m) denote the maximal number of distinct rows in any (0, 1 )-matrix with n columns, rank < n, – 1, and all row sums equal to m. This paper determines g(n, m) in all cases:In addition, it is shown that if V is a k-dimensional vector subspace of any vector space, then V contains at most 2k vectors all of whose coordinates are 0 or 1.

scholarly journals The expected dimension of a sum of vector subspaces

1992 ◽  
Vol 45 (3) ◽  
pp. 467-477 ◽  
Author(s):  
David E. Dobbs ◽  
Mark J. Lancaster
Keyword(s):  
Vector Space ◽  
Vector Subspace ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Limiting Value ◽  
Vector Subspaces

Let W be an n−dimensional vector space over a field F. It is shown that the expected dimension of a vector subspace of W is n/2. If F is infinite, the expected dimension of a sum of a pair of subspaces of W is (n + 1)/2 if n > 1; and 3/4 if n = 1. If F is finite, with q elements, the expected dimension of a sum of subspaces of W depends on q and n. For fixed n, the limiting value of this expectation as q → ∞ is n if n is even; and n − 1/4 if n is odd. Moreover, if F is finite and n > 1, the expected dimension of a sum of three (not necessarily distinct) subspaces of W has limit n as q → ∞.

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

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Efficient Approach ◽  
Transfer Homomorphism ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with $\deg(x_j) = 1.$ Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It is called \textit{the "hit" problem} for Steenrod algebra and has been completely solved for $t\leq 4.$ In this article, we explicitly solve the hit problem of five variables in the "generic" degree $n=5(2^{s}-1) + 42.2^{s}$ for every non-negative integer $s.$ The result confirms Sum's conjecture [15] for the relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and degree $n$ above. An efficient approach for surveying the hit problem of five variables has been presented. As an application, we obtain the dimension of $(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n$ for $t = 6$ and degree $5(2^{s+5}-1) + 42.2^{s+5}$ for all $s\geq 0.$ At the same time, we show that the Singer transfer homomorphism is an isomorphism in bidegree $(5, 5+n)$.

$\mathscr A$-generators for the polynomial algebra of five variables in degree $5(2^t - 1) + 6.2^t$

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Polynomial Algebra ◽  
Minimal System ◽  
Steenrod Algebra ◽  
Arbitrary Positive Integer ◽  
Left Module ◽  
Minimal System Of Generators ◽  
Set Up ◽  
Hit Problem

Let $P_s:= \mathbb{F}_2[x_1,x_2,\ldots ,x_s] = \bigoplus_{n\geqslant 0}(P_s)_n$ be the polynomial algebra viewedas a graded left module over the mod 2 Steenrod algebra, $\mathscr A.$ The grading is by the degree of the homogeneous terms $(P_s)_n$ of degree $n$ in the variables $x_1, x_2, \ldots, x_s$ of grading $1.$ We are interested in the {\it hit problem}, set up by F.P. Peterson, of finding a minimal system of generators for $\mathscr A$-module $P_s.$ Equivalently, we want to find a basis for the $\mathbb F_2$-graded vector space $\mathbb F_2\otimes_{\mathscr A} P_s.$ In this paper, we study the hit problem in the case $s=5$ and the degree $n = 5(2^t-1) + 6.2^t$ with $t$ an arbitrary positive integer.

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

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Vector Space ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Central Problem ◽  
Minimal Set ◽  
Prime Field ◽  
Efficient Approach ◽  
Minimal Sets ◽  
Hit Problem

We denote by $\mathbb Z_2$ the prime field of two elements and by $P_t = \mathbb Z_2[x_1, \ldots, x_t]$ the polynomial algebra of $t$ generators $x_1, \ldots, x_t$ with $\deg(x_j) = 1.$ Let $\mathcal A_2$ be the Steenrod algebra over $\mathbb Z_2.$ A central problem of homotopy theory is to determine a minimal set of generators for the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It is called \textit{the "hit" problem} for Steenrod algebra and has been completely solved for $t\leq 4.$ In this article, we explicitly solve the hit problem of five variables in the generic degree $5(2^{s}-1) + 42.2^{s}$ for any $s\geq 0.$ The result confirms Sum's conjecture \cite{N.S2} for the relation between the minimal sets of $\mathcal A_2$-generators of the algebras $P_{t-1}$ and $P_{t}$ in the case $t=5$ and the above generic degree. An efficient approach to surveying the hit problem of five variables has been presented. As an application, we obtain the dimension of $(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n$ for $t = 6$ and the generic degree $n = 5(2^{s+5}-1) + 42.2^{s+5}$ for all $s\geq 0.$ At the same time, we show that the fifth Singer algebraic transfer is an isomorphism in bidegree $(5, 47.2^{s})$ with $s\geq 0.$

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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