Sparseness for the symmetric hit problem at all primes

Pengelley, David; Williams, Frank

doi:10.1017/s0305004114000668

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…The Dickson algebra is also an unstable A 2 -module and is dual to the coalgebra of Dyer-Lashof operations of the length d (see Madsen [25]). The relationship between Kudo-Araki-May algebra and the hit problem has been investigated by Pengelley and Williams [32,34,36], and by Singer [57]. In [5], Ault and Singer have examined the dual problem of the Peterson hit problem, which is to determine the set of A + 2 -annihilated elements in the homology of B(Z/2) ×d .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Phúc¹

2021

Preprint

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: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Phúc¹

2021

Preprint

103

View full text Add to dashboard Cite

show abstract

“…The Dickson algebra is also an unstable A 2 -module and is dual to the coalgebra of Dyer-Lashof operations of the length d (see Madsen [25]). The relationship between Kudo-Araki-May algebra and the hit problem has been investigated by Pengelley and Williams [32,34,36], and by Singer [59]. In [5], Ault and Singer have examined the dual problem of the Peterson hit problem, which is to determine the set of A + 2 -annihilated elements in the homology of B(Z/2) ×d .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Phúc¹

2021

Preprint

View full text Add to dashboard Cite

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 $\mathbb Z/2[x_1, x_2, \ldots, x_d],$ which is viewed as a connected unstable $\mathcal A_2$-module on $d$ generators of degree one. We study the Peterson "hit problem" of finding the minimal set of $\mathcal A_2$-generators for $\mathscr P_d.$ It is equivalent to determining a $\mathbb Z/2$-basis for the space of "cohits"$$Q\mathscr P_d := \mathbb Z/2\otimes_{\mathcal A_2} \mathscr P_d \cong \mathscr P_d/\mathcal A_2^+\mathscr P_d.$$ This $Q\mathscr P_d$ is considered as a form modular representation of $GL_d$ over $\mathbb Z/2.$ The problem for $d= 5$ is not yet completely solved, and unknown 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 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 mysterious Ext groups and the Kervaire invariant one problem.

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“…Since its formulation in mid-1980 by Peterson through the computation of QP d 2 , the hit problem has been and is studied by many mathematicians. Among recently published papers and books are Ault [1], Pengelley and Williams [4], and Sum [6] and Walker and Wood [8].…”

Section: Introductionmentioning

confidence: 99%