A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

Pengelley, David; Williams, Frank

doi:10.2140/agt.2011.11.1767

Cited by 8 publications

(9 citation statements)

References 21 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

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“…Our primary tool in this description will be the self-map of S (whose definition we shall recall in Section 2) given by the element d 2 2 K, the Kudo-Araki-May algebra. As in [6] we shall see that S is a free module over d 2 , and we shall solve the problem of computing S by finding a d 2 -basis for it. A key ingredient is that for p 2 the map d 2 W S !…”

Section: Discussionmentioning

confidence: 99%

“…Recall [6] that for any prime p , H .BUI F p / is the polynomial algebra with generators a n 2 H 2n .BUI F p / for n 1, dual to the powers c n 1 of the first Chern class, and that H .BU.l/I F p / can be thought of as the subspace spanned by monomials in the a n of length at most l . It is convenient for us, and is usual in the literature, to introduce a placeholder, a 0 , of topological degree zero, so that a monomial the action for CP .1/ D BU.1/, in which a 0 is both primitive and never hit by a positive operation, ie, transparent to the A-action.…”

Section: The A-action On Mmentioning

confidence: 99%

“…Recall [6] that for any prime p , the action of the element d 2 2 K on a i a j 2 H .BUI F p / was defined by the formula d 2 .a i a j / D a piCp 1 a pj Cp 1 for i; j 1, and we extend this definition to i; j 0. Kameko [5] and Singer [8] initiated the use of similar operations at the prime 2 for the hit problem, and these have been motivational for our work [6]. It is easy to check that (2) .…”

Section: Booting With D 2 To Organize Primitivesmentioning

confidence: 99%

“…The problem has been studied at the prime p D 2 for polynomial algebras with generators in degree one (cohomology of products of projective spaces), and more recently for algebras of symmetric polynomials in such generators, which are the cohomologies of the classifying spaces BO.l/. The hit problem for various classifying spaces and primes has received considerable attention, and partial results have been obtained in Crossley [1; 2], Janfada and Wood [3; 4], Kameko [5], Pengelley and Williams [6], Peterson [7], Singer [8] and Wood [9]. We refer to [6] for further background.…”

Section: Introductionmentioning

confidence: 99%

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