Lambda algebra and the Singer transfer

Chơn, Phan Hoàng; Hà, Lê Minh

doi:10.1016/j.crma.2010.11.008

Cited by 53 publications

(101 citation statements)

References 16 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: 64%

“…We refer to Sect.2 for the concept of the admissible monomial.) Using this result combining with the computations of Ext 5,13.2 t A2 (Z/2, Z/2) (see Tangora [66], Chen [10], Lin [23]), and a direct computation using a result in [11] on the representation in the Z/2-lambda algebra Λ of the transfer homomorphism of rank 5, we show that T r 5 is an isomorphism when acting on Z/2 ⊗ GL5 P A2 H 13.2 t −5 (B(Z/2) ×5 ) for t ∈ {0, 1}. (The information on the algebra Λ can be found below in this section.)…”

Section: A2mentioning

confidence: 93%

“…(Note that Sq k * denotes the dual of Sq k .) In [11], Chơn and Hà have established a homomorphism ψ d :…”

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: 64%

Section: A2mentioning

confidence: 93%

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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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“…Recall that Λ is the quotient of the graded tensor algebra over Z/2 on symbols λ i for i ≥ −1, modulo the two-sided ideal generated by λ s λ k − j j−k−1 2j−s λ s+k−j λ j , for any s, k ≥ −1 by the right ideal generated by λ −1 . An interesting representation in the algebra Λ of the algebraic transfer, established by Chơn and Hà [5], is a Z/2-linear map ψ n from P A H * (V ⊕n ) to a subspace of Λ spanned by all monomials of length n in all the monomials in λ i . The authors showed that the image of an element…”

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confidence: 99%

On the lambda algebra and Singer's cohomological transfer

Phúc¹

2021

Preprint

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Write $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in \cite{Singer}, which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb A.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

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“…It is a useful tool in describing the homology groups of the Steenrod algebra, Tor A k,k+d (F 2 , F 2 ). This transfer was studied by Boardman [2], Bruner, Ha and Hung [3], Ha [5], Hung [9], Chon and Ha [4,10,11], Minami [12], Nam [7], Hung and Quynh [6], the present author [13] and others.…”

Section: Introductionmentioning

confidence: 99%

On the determination of the Singer transfer

Sum¹

2018

VJSTE

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Abstract:Let P k be the graded polynomial algebra F 2 [x 1 , x 2 , . . . , x k ] with the degree of each generator x i being 1, where F2 denote the prime field of two elements, and let GL k be the general linear group over F 2 which acts regularly on P k .We study the algebraic transfer constructed by Singer [1] using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra A, Tor

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Lambda algebra and the Singer transfer

Cited by 53 publications

References 16 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

On the lambda algebra and Singer's cohomological transfer

On the determination of the Singer transfer

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