On the determination of the Singer transfer

Sum, Nguyen

doi:10.31276/vjste.60(1).03

Cited by 15 publications

(48 citation statements)

References 20 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: 65%

“…Indeed, by using a computer calculation, Hưng provided a counter-example in [19] that Sq 0 is not a monomorphism when acting on Z/2 ⊗ GL5 P A2 H 15 (B(Z/2) ×5 ). This was confirmed again by the works of Sum [61,65]. Thereafter, Hưng [19] conjectured that Sq 0 is a monomorphism if and only if d ≤ 4.…”

Section: Preliminariesmentioning

confidence: 69%

“…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%

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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: 65%

Section: Preliminariesmentioning

confidence: 69%

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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“…This means that k ⊗ GL 4 (k) P ((P 4 ) * 2(2 1 −1)+2 1 ) is trivial. For s ∈ {2, 4}, combining Theorems 2.2, 2.9 with the inequality (8) and the fact that the invariant space (Q ⊗4 2 2−1 +2 2 −3 ) GL 4 (k) is trivial (see Sum [44]), it may be concluded that the coinvariant spaces k ⊗ GL 4 (k) P ((P 4 ) * 2(2 s −1)+2 s ) are trivial, too. For s ∈ {1, 2, 4}, the following inequality is immediate from Theorems 2.2 and 2.9 and the inequality ( 8): (10) dim k ⊗ GL 4 (k) P ((P 4 ) * 2(2 s −1)+2 s ) ≤ 1.…”

Section: Theorem 25 With a Positive Integer S We Havementioning

confidence: 99%

“…where P ((P q ) * n ) := {θ ∈ (P q ) * n : (θ)Sq i = 0, for all i > 0} = (Q ⊗q n ) * , the space of primitive homology classes as a representation of GL q (k) for all n and the coinvariant k⊗ GLq(k) P ((P q ) * n ) is isomorphic as an k-vector space to (Q ⊗q n ) GLq(k) , the subspace of GL q (k)-invariants of Q ⊗q . The Singer transfer has been studying for a long time: see Boardman [4], Chơn and Hà [9], Crossley [10], Hà [15], Hưng [18], Hưng-Quỳnh [19], Minami [26], the present writer [31,33,34,35,36,37], Sum [44,46], and others. By the works of Singer himself [39] and Boardman [4], T r A q is known to be an isomorphism for q ≤ 3.…”

Section: Introductionmentioning

confidence: 99%

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} \mathscr P_AH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

Phúc¹

2021

Preprint

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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 $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This is the content of "hit problem" of Frank Peterson. Based on this problem, we are interested in the $q$-th algebraic transfer $Tr_q^{A}$ of W. Singer \cite{W.S1}, which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a linear map from the space of $GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field $k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ The present paper is to investigate this algebraic transfer for the cohomological degree $q = 4.$ More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and a representation of the rank 4 transfer over the lambda algebra, we show that $Tr_4^{A}$ is an isomorphism in respective degrees. Also, we give some conjectures on the dimensions of $k\otimes_{GL_q(k)} ((P_4)_n^{*})$ for the remaining degrees $n.$ As a consequence, Singer's conjecture for the algebraic transfer is true in the rank 4 case. This study and our previous results \cite{D.P11, D.P12} provided a complete picture of the behavior of $Tr_4^{A}.$

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On a Construction for the Generators of the Polynomial Algebra as a Module Over the Steenrod Algebra

Sum

2019

Trends in Mathematics

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On the determination of the Singer transfer

Cited by 15 publications

References 20 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

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb Z_2\otimes_{GL_4(\mathbb Z_2)} \mathscr P_AH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some generic degrees and its application

On a Construction for the Generators of the Polynomial Algebra as a Module Over the Steenrod Algebra

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