Polynomials and the mod 2 Steenrod Algebra

Walker, Grant; Wood, Reginald M. W.

doi:10.1017/9781108304092

Cited by 18 publications

(42 citation statements)

References 110 publications

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“…In Sect.5, we compile a comprehensive list of all the allowable monomials of degree n 0 = 5(2 0 − 1) + 18.2 0 , and also enumerate the strictly non-allowable monomials of degree n 1 = 5(2 1 − 1) + 18.2 1 contained within the A -module P 5 . Additionally, we also show in the Appendix that a local version of Kameko's conjecture as informed in [WW18b] holds true for any m ≥ 1 and parameter vectors of degrees ≤ 12.…”

Section: Theorem 21 Given the Arithmetic Functionmentioning

confidence: 70%

“…The classical "hit problem" for the algebra P m , which is concerned with seeking a minimal set of A -generators for P m , has been initiated in a variety of contexts by Peterson [Pet87], Priddy [Pri90], Singer [Sin91], and Wood [Woo89]. The study of modules over the Steenrod algebra A and related hit problems is a central topic in Algebraic Topology and has received extensive attention from many mathematicians, including Ault and Singer [AS11], Brunetti, Ciampella and Lomonaco [BCL12], Brunetti and Lomonaco [BL14], Crabb and Hubbuck [CH96], Janfada [Jan08], Kameko [Kam90], Mothebe et al [MU15,Mot16,MKR16], Nam [Nam04], Repka and Selick [RS98], Walker and Wood [WW18a,WW18b], the present author [Phu16, Phu20a, Phu20b, Phu21a, Phu21b, Phu22a, Phu22b, Phu22c, Phu22d, Phu23a, Phu23d], etc. We believe that the best references here are the books by Walker and Wood [WW18a,WW18b], where the most popular articles on these subjects are collected.…”

Section: Context and Overviewmentioning

confidence: 99%

“…That is why we became interested in programming and exploring easier ways to find a solution to the general problem. On the other vein, inspired by some noteworthy findings in the literature [Sil98,WW18a,WW18b], Sum [Sum23] has recently proposed an alternative approach to determine the dimension of the kernel of the Kameko ( Sq 0 * ) (m,n) in a specific type of generic degrees n. Honestly, his technique is very complicated, and not everyone will be able to grasp it. However, it offers us a more multi-dimensional viewpoint and the prospect of discovering a comprehensive solution to the hit problem.)…”

Section: Theorem 21 Given the Arithmetic Functionmentioning

confidence: 99%

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On modules over the mod 2 Steenrod algebra and hit problems

Phúc¹

2023

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<p>Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ The classical "hit problem" in Algebraic Topology, which is widely recognized as an important and intriguing open problem, asks for a minimal set of generators for the polynomial algebra, $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, on $m$ variables $x_1, \ldots, x_m$, each of which has degree one, regarded as a connected unstable $\mathscr A$-module. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1)$. Despite extensive study over the past three decades, the hit problem remains unresolved for $m\geq 5$. </p> <p>In this article, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on the hit problem for $\mathscr A$-module $\mathcal P_5$ in generic degree $n_s = 5(2^{s}-1) + 18.2^{s}$ with $s$ an arbitrary non-negative integer. As a result, a local version of Kameko's conjecture, which concerns the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ in relation to weight vectors, has been confirmed for the case where $m = 5$ and the degree is $n_s.$ Also, we show that this conjecture also holds for any $m\geq 1$ and degrees $\leq 12.$ This study has two important applications: (1) it establishes the dimension result for the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ for $m = 6$ in generic degree $5(2^{s+4}-1) + n_1.2^{s+4}$ with $s > 0;$ and (2) it describes the representations of the general linear group of rank $5$ over $\mathbb F_2.$ As a result, we prove that the algebraic transfer, defined by W. Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Besides, we obtain new results on the behavior of this algebra transfer for all homological degrees $m$. Specifically, we demonstrate that Singer's transfer is a trivial isomorphism in bidegree $(m, m+12)$ for any $m > 0$.</p>

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Section: Theorem 21 Given the Arithmetic Functionmentioning

confidence: 70%

Section: Context and Overviewmentioning

confidence: 99%

Section: Theorem 21 Given the Arithmetic Functionmentioning

confidence: 99%

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On modules over the mod 2 Steenrod algebra and hit problems

Phúc¹

2023

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“…Note that this theorem has been proved by Mothebe [7] for s ¼ 0, and by Walker and Wood [26] for s ¼ 1, and by another method.…”

Section: <mentioning

confidence: 80%

“…Then, this problem was investigated by Wood [28], Carlisle and Wood [1], Silverman [17], Nam [8,9], Mothebe [7], Sum [19,21], Cho'n and Hà [3], Kameko [5,6] and others. Recently, the hit problem and its applications to representations of general linear groups have been presented in the books of Walker and Wood [26,27].…”

Section: > < >mentioning

confidence: 99%

A note on the Peterson hit problem for the Steenrod algebra

Tin¹

2021

Proc. Japan Acad. Ser. A Math. Sci.

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x 2 ; . . . ; x k be the graded polynomial algebra over the prime field of two elements F 2 , in k generators x 1 ; x 2 ; . . . ; x k , each of degree 1. Being the mod-2 cohomology of the classifying space BðZ=2Þ k , the algebra P k is a module over the mod-2 Steenrod algebra A.In this Note, we explicitly compute the hit problem of some generic degrees rð2 s À 1Þ þ 2 s m in P k , where r ¼ k À 1 ¼ 4; m 2 f8; 10; 11g 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 ¼ 5 and 6.

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On the hit problem for the Steenrod algebra in the generic degree and its applications

Tin

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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Polynomials and the mod 2 Steenrod Algebra

Cited by 18 publications

References 110 publications

On modules over the mod 2 Steenrod algebra and hit problems

On modules over the mod 2 Steenrod algebra and hit problems

A note on the Peterson hit problem for the Steenrod algebra

On the hit problem for the Steenrod algebra in the generic degree and its applications

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