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

Abstract

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.

Structure of the space of GL4 (Z2)-coinvariants Z2 ⊗ GL4 (Z2) P H∗ (Z42,Z2) in some generic degrees and its application

“…Interested readers can find illustrative counterexamples in [34,43]. We suggest that readers refer to recent works, such as [23,24,25,26,28,29,40], for comprehensive information on the solutions of the hit problem for variables greater than 4 in certain "generic" degrees. Additionally, in [30], our initial study yielded new results on the symmetric hit problem for the symmetric polynomial algebra in four variables, P Σ 4 4 .…”

“…This transfer is a k-linear map T r A q : k ⊗ G(q) P A ((P q ) * n ) −→ Ext dim k q ,dim k q +n A (k, k) = Ext q,q+n A (k, k), on which P A ((P q ) * n ) := {θ ∈ (P q ) * n : (θ)Sq i = 0, for all i > 0} = (Q Q Q q n ) * denotes the space of primitive homology classes as a representation of G(q) for all n, and the coinvariant k⊗ G(q) P A ((P q ) * n ) is isomorphic as an k-vector space to (Q Q Q q n ) G (q) , the subspace of G(q)-invariants of Q Q Q q n . Singer's transfer has been studied by many topologists like Boardman [3], Bruner et al [5], Chơn and Hà [7,8], Hà [10], Hưng [12], Hưng and Quỳnh [13], Minami [19], Nam [21], the present writer [23,24,25,27,28], etc. However, it is also not a straightforward task to explicitly understand the structure of the (co)domain of the transfer.…”

On a minimal set of generators for the algebra H ∗ ( B E d ; F 2 ) and its applications

“…Consider the degree m r := 6(2 r − 1) + 6.2 r , for r = 2. By using the MAGMA computer algebra system, Phuc showed in [10] that the F 2 -vector space (AP 5 ) 42 has 2520-dimensional (see [10], pp.4), where dim(AP 0 5 ) 42 = 700, and dim(AP + 5 ) 42 = 1820. Assume that the set e i ∈ (P 5 ) 42 : 1 i 2520 is a minimal set of generators for A-modules P 5 in degree forty-two.…”

The affirmative answer to Singer's conjecture on the algebraic transfer of rank four

“…This transfer homomorphism has been the subject of extensive research and has been found to be highly non-trivial. Many studies have been conducted to better understand it, including works by Boardman [4], Minami [15], Bruner, Hà, and Hưng [7], Hưng [11], Hà [10], Nam [17], Hưng and Quỳnh [12], Chơn, and Hà [9], Sum [35], the author [19,20,21,22,23,24,25,26,27,28], and others. To provide the reader with a comprehensive understanding of this Singer transfer, we will consider the associated objects.…”

“…The hit problem has received extensive attention from numerous topologists for over three decades, as evidenced by references including [4], [7], [13], [19,20,21,22,24,27,28], [32,33,34,36], [38], [42].…”