Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON PETERSON'S OPEN PROBLEM AND REPRESENTATIONS OF THE GENERAL LINEAR GROUPS

  • Phuc, Dang Vo (Faculty of Education Studies University of Khanh Hoa)
  • Received : 2020.04.25
  • Accepted : 2020.12.09
  • Published : 2021.05.01
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Abstract

Fix ℤ/2 is the prime field of two elements and write 𝒜2 for the mod 2 Steenrod algebra. Denote by GLd := GL(d, ℤ/2) the general linear group of rank d over ℤ/2 and by ${\mathfrak{P}}_d$ the polynomial algebra ℤ/2[x1, x2, …, xd] as a connected unstable 𝒜2-module on d generators of degree one. We study the Peterson "hit problem" of finding the minimal set of 𝒜2-generators for ${\mathfrak{P}}_d$. Equivalently, we need to determine a basis for the ℤ/2-vector space $$Q{\mathfrak{P}}_d:={\mathbb{Z}}/2{\otimes}_{\mathcal{A}_2}\;{\mathfrak{P}}_d{\sim_=}{\mathfrak{P}}_d/{\mathcal{A}}^+_2{\mathfrak{P}}_d$$ in each degree n ≥ 1. Note that this space is a representation of GLd over 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(2t - 1) + 2ts 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 + (13.20 - 5)) and (5, 5 + (13.21 - 5)). Moreover, the result when t ≥ 2 was also discussed. Here, the Singer transfer of rank d is a ℤ/2-algebra homomorphism from GLd-coinvariants of certain subspaces of $Q{\mathfrak{P}}_d$ to the cohomology groups of the Steenrod algebra, $Ext^{d,d+*}_{\mathcal{A}_2}$ (ℤ/2, ℤ/2). It is one of the useful tools for studying these mysterious Ext groups.

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References

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