A representation of the dual of the Steenrod algebra

Brunetti, Maurizio; Lomonaco, Luciano Amito

doi:10.1007/s11587-014-0191-y

Cited by 20 publications

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“…The lemma shows that Ker(( Sq 0 * ) 53 ) = QP ⊗5 53 (3,3,3,2,1). By similar calculations as the case s = 0, we find that the invariant (QP ⊗5 53 (3, 3, 3, 2, 1)) GL 5 = 0, and so (Ker(( Sq 0 * ) 53 )) , belongs to H n s ((RP (∞) 5 ) for all s ≥ 1. Combining this together with the case s = 0, we may conclude that (Z/2 ⊗ GL 5 P A (P ⊗5 ) * )…”

Section: Lemma 3115 If T ∈ C ⊗5supporting

confidence: 57%

“…We see that the cardinality of this set is 5 k . For each J ∈ L 5,k , we define the the homomorphism ϕ J : P ⊗k −→ P ⊗5 of algebras by substituting ϕ J (t u ) = t j u , 1 ≤ u ≤ 5.…”

Section: Notations and Preliminariesmentioning

confidence: 93%

“…space, called Steenrod operations of degrees k. They are stable cohomology operations, that is, they commute with suspension maps. The Steenrod operations and related problems have been investigated by many authors (for instance Brunetti, Ciampella, and Lomonaco [4], Brunetti, and Lomonaco [5], Nicholas [21]). All these Sq k form an algebraic structure which is known as the 2-primary Steenrod algebra A subject to the Adem relations (see also Walker and Wood [44]).…”

mentioning

confidence: 99%

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On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables

Phúc¹

2022

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Let us consider the Steenrod algebra $\mathcal A$ over the field of two elements, $\mathbb Z/2.$ We knew that that the $\mathbb Z/2$-cohomology of the product of $h$ copies of the infinite real projective space $\mathbb RP(\infty)$ can be identified with $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h],$ the polynomial algebra on $h$ generators with the degree of each $t_i$ being one. Moreover, this $P^{\otimes h}$ equipped with the (left) unstable $\mathcal A$-module structure. In the work [Abstracts Amer. Math. Soc. 833 (1987)], Franklin Peterson proposed the "hit" problem of determination a minimal generating set for $\mathcal A$-module $P^{\otimes h}.$ Equivalently, the hit problem is to find a monomial basis for the unhit space $\mathbb Z/2\otimes_{\mathcal A}P^{\otimes h}$ in each positive degree. The hit problem, which remains open for arbitrary $h\geq 5,$ plays an important role in discovering some classical problems in homotopy theory. Singer's algebraic transfer of rank $h$ [Math. Z. 202, 493-523 (1989)], which passes from the coinvariants of certain representation of the general linear group $GL_h$ of rank $h$ over $\mathbb Z/2$ to the $\mathbb Z/2$-cohomology group of the Steenrod algebra, ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb Z/2, \mathbb Z/2),$ is a relatively efficient tool to describe mysterious Ext groups. Singer conjectured that this transfer homomorphism is a monomorphism, but there is no answer yet in ranks $\geq 5.$ It is very difficult to calculate the value of the algebraic transfer on any non-zero element in each positive degree. In the current paper, we use techniques of the hit problem for the algebra $\mathcal A$ to study the algebraic transfer. The approach is quite effective to determine the Singer transfer. Our result then shows that the algebraic transfer of rank 5 in the general degree given is an isomorphism. As a consequence, Singer's conjecture is true for rank $5$ in those degrees.

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Section: Lemma 3115 If T ∈ C ⊗5supporting

confidence: 57%

Section: Notations and Preliminariesmentioning

confidence: 93%

mentioning

confidence: 99%

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On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables

Phúc¹

2022

Preprint

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“…By Cartan's formula, it suffices to determine Sq i (t j ) and the instability axioms give Sq 1 (t j ) = t 2 j while Sq k (t j ) = 0 if k > 1. The Steenrod operations and related problems have been investigated by many authors (for instance Brunetti, Ciampella and Lomonaco [6], Brunetti and Lomonaco [7], Singer [41], Silverman [39], Monks [17], Wood [54]). The Steenrod algebra naturally acts on the cohomology ring H * (X) of a CW-complex X.…”

Section: Introductionmentioning

confidence: 99%

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Phúc

2023

Journal of Algebra

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“…By Cartan's formula, it suffices to determine Sq i (t j ) and the instability axioms give Sq 1 (t j ) = t 2 j while Sq k (t j ) = 0 if k > 1. The Steenrod operations and related problems have been investigated by many authors (for instance Brunetti, Ciampella and Lomonaco [6], Brunetti and Lomonaco [7], Singer [42], Silverman [40], Monks [18], Wood [56]). The Steenrod algebra naturally acts on the cohomology ring H * (X) of a CW-complex X.…”

Section: Introductionmentioning

confidence: 99%

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Phúc¹

2022

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<p>Let $\mathcal A$ be the classical, singly-graded Steenrod algebra over the prime order field $\mathbb F_2$ and let $P^{\otimes h}: = \mathbb F_2[t_1, \ldots, t_h]$ denote the polynomial algebra on $h$ generators, each of degree $1$, viewed as a module over $\mathcal A.$ Write $GL_h$ for the usual general linear group of rank $h$ over $\mathbb F_2.$ As well known, the (mod 2) cohomology groups of the Steenrod algebra, ${\rm Ext}_{\mathcal A}^{h, h+*}(\mathbb F_2, \mathbb F_2)$ is still largely mysterious for all homological degrees $h \geq 6.$ The $h$-th algebraic transfer $$Tr_h^{\mathcal A}: (\mathbb F_2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}]^{*})_n\longrightarrow {\rm Ext}_{\mathcal A}^{h, h+n}(\mathbb F_2, \mathbb F_2),$$ defined by William Singer \cite{Singer}, is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any $h\geq 5.$ There is currently no information on the conjecture for $h = 6$. In this paper, we verify Singer's conjecture for all homological degrees $h\geq 1$ in the internal degrees $\leq 10 = 6(2^{0}-1) + 10\cdot 2^{0}$ and for $h = 6$ in the degree of the general form $n_s:=6(2^{s}-1) + 10\cdot 2^{s}.$ This result is important, since it tells us that the non-zero elements $h_2^{2}g_1 = h_4Ph_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_1}(\mathbb F_2, \mathbb F_2)$, and $D_2\in {\rm Ext}_{\mathcal A}^{6, 6+n_2}(\mathbb F_2, \mathbb F_2)$ are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one \cite{Phuc11}, which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case $h = 6$ and degree $n_s.$ This extends a result in Mothebe, Kaelo and Ramatebele \cite{MKR}. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, $\mathbb F_2\otimes_{\mathcal A}P^{\otimes 7}$ in a certain general degree. </p>

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A representation of the dual of the Steenrod algebra

Abstract: In this paper we show how to embed A∗, the dual of the mod 2 Steenrod\ud algebra, into a certain inverse limit of algebras of invariants of the general linear group.\ud The prime 2 is fixed throughout the pape

Cited by 20 publications

References 4 publications

On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables

On the modular representations of the general linear group $GL_5$ and the hit problem for the polynomial algebra of five variables

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

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