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• W2962787875 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 $mathbb Z/2[x_1, x_2, ldots, x_d]$ as a connected unstable $mathcal A_2$-module on $d$ generators of degree one. We study the Peterson 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 $Qmathscr P_d := mathbb Z/2otimes_{mathcal A_2} mathscr P_d cong mathscr P_d/mathcal A_2^+mathscr P_d.$ This $Qmathscr P_d$ is also a 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 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 $tgeq 2$ 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 $Qmathscr P_d$ 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." @default.
• W2962787875 created "2019-07-30" @default.
• W2962787875 creator A5018079701 @default.
• W2962787875 date "2019-07-20" @default.
• W2962787875 modified "2023-09-27" @default.
• W2962787875 title "On Peterson's open problem and representations of the general linear groups" @default.
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• W2962787875 hasPublicationYear "2019" @default.
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