Some Relations between Admissible Monomials for the Polynomial Algebra

Mothebe, Mbakiso Fix; Uys, Lafras

doi:10.1155/2015/235806

Cited by 48 publications

(59 citation statements)

References 13 publications

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“…This is demonstrated with the aid of known results for dim(Q d (5)) (cited in Table A). The results also show an improvement from the results obtained in (Mothebe and Uys, 2015) by application of Theorem 3.…”

Section: Given Two Sequencessupporting

confidence: 61%

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

Mothebe¹

2016

JMR

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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 $n=5.$

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Section: Given Two Sequencessupporting

confidence: 61%

“…Theorem 2 is a generalization of the following result of the author and Uys proved in (Mothebe and Uys, 2015).…”

Section: Given Two Sequencesmentioning

confidence: 72%

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

Mothebe¹

2016

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“…From this event, one of the applications of the hit problem of Peterson is to study the representations of the general linear groups over Z/2. Therefrom the hit problem has attracted great interest of many algebraic topologists (see Crabb and Hubbuck [12], Crossley [13], Kameko [21], Mothebe and his collaborators [27,28,29], Nam [30], Pengelley and Williams [33,35], Priddy [52], Silverman and Singer [54], Singer [56], Peterson [38], the present author [39,40,41,42,43,44,50,51], Sum [59,60,61,62,63,64,65], Walker and Wood [69,70,71], Wood [74,75] and others).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Phúc¹

2021

Preprint

103

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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.

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“…Since x ∈ C ⊗5 42 , according to Theorem 2.5, y ∈ C ⊗5 19 . Then, by Tín [16], we have ω(y) ∈ {(3, 2, 1, 1), (3,2,3), (3,4,2)}. This implies that ω(x) is one of the sequences ω (j) , 1 ≤ j ≤ 3.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…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. )…”

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confidence: 99%

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

Phúc¹

2021

Preprint

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One of the open problems in Algebraic topology is the Peterson hit problem for the algebra polynomial $P_t = \mathbb Z_2[x_1, \ldots, x_t] = H^{*}(B\mathbb Z_2^{t}, \mathbb Z_2)$ of $t$ generators $x_1, \ldots, x_t$ with $|x_j| = 1.$ The problem seeks a minimal set of the $\mathbb Z_2$-graded vector space $\{(\mathbb Z_2\otimes_{\mathcal A_2} P_t)_n\}_{n\geq 0}.$ It has been thoroughly studied for 35 years in a variety of contexts by many authors and completely solved for $t\leq 4.$ In the case $t = 5$ and some "generic" degrees, the problem was studied by many authors. 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} on the relation between the minimal set of $\mathcal A_2$-generators for 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 in this case 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 $(5, 47.2^{s})$ for any $s\geq 0.$

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Some Relations between Admissible Monomials for the Polynomial Algebra

Cited by 48 publications

References 13 publications

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

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

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

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

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