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

Phúc, Đặng Võ

doi:10.1016/j.topol.2020.107321

Cited by 59 publications

(211 citation statements)

References 26 publications

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“…al. [9], Chơn and Hà [11], Crossley [14], Hà [15], Hưng [19], Minami [26], Nam [31], the present author [41,42,43,44,46,47,48,49,50,51], Sum [61,62,63,65] and others). In [55], using the invariant theory, Singer claims that T r d is an isomorphism for d = 4 in a range of internal degrees, but T r 5 is not an epimorphism.…”

Section: A2mentioning

confidence: 63%

“…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%

“…Recall that to solve the hit problem of Peterson, we will determine QP d in each degree n ≥ 1. However, as explicitly shown in [43], it is enough to consider this space in the following "generic degree":…”

Section: A2mentioning

confidence: 99%

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

Phúc¹

2021

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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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Section: A2mentioning

confidence: 63%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Phúc¹

2021

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103

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“…However, when s 5, it is an open problem. Recently, some authors have been studied the conjecture for s = 4, 5 (see Bruner-Hà-Hưng [4], Hưng [11], Chơn-Hà [7,8], Hà [9], Nam [22], the present author [26], [28]- [35], Sum [45,47,48,49] and others). In the present work, by using techniques of the hit problem of five variables, we investigate Conjecture 1.2 in bidegree (5, d + 5), where d is determined as in Theorem 1.1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We explicitly compute p (i;I) (S) in terms u j , 1 j 23. By a direct computation using Lemma 2.2.9,Theorem 3.1.3, and from the relations p (i;j) (S) ≡ ω (5,2) 0 with either i = 1, j = 2, 3 or i = 2, j = 3, 4, one gets Here J = {1, 2, 3,4,5,6,7,8,9,10,11,12,15,16,17,18,19,21,22,23,24,25,26,28,29,31,38,39,40,41,42,45,46,48,50,51,57 [46]). By a simple computation, we get…”

Section: We Now Prove the Setmentioning

confidence: 99%

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

Phúc¹

2021

Preprint

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Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree $1$. We are interested in the {\it Peterson "hit" problem} of finding a minimal set of generators for $P_s$ as a graded left module over the mod-2 Steenrod algebra, $\mathscr {A}$. For $s\geqslant 5,$ it is still open.In this paper, we study the hit problem of five variables in a generic degree. By using this result, we survey Singer's conjecture for the fifth algebraic transfer in the respective degrees. This gives an efficient method to study the algebraic transfer and it is different from the ones of Singer

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On $$\mathbb A$$-generators of the cohomology $$H^{*}(V^{\bigoplus 5}) = \mathbb Z/2[u_1, \ldots , u_5]$$ and the cohomological transfer of rank 5

Phúc

2023

Rend. Circ. Mat. Palermo, II. Ser

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

Cited by 59 publications

References 26 publications

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

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

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

On $$\mathbb A$$-generators of the cohomology $$H^{*}(V^{\bigoplus 5}) = \mathbb Z/2[u_1, \ldots , u_5]$$ and the cohomological transfer of rank 5

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