A note on the generators of the polynomial algebra of six variables and application

Tin, Nguyen Khac

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Let \mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}] be the graded polynomial algebra over \mathcal K, where \mathcal K denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra \mathcal P_{n}, viewed as a graded left module over the mod-2 Steenrod algebra, \mathcal{A}. For n>4, this problem is still unsolved, even in the case of n=5 with the help of computers.

In this paper, we study the hit problem for the case n=6 in degree d_{k}=6(2^{k} -1)+9.2^{k}, with k an arbitrary non-negative integer. By considering \mathcal K as a trivial \mathcal A-module, then the hit problem is equivalent to the problem of finding a basis of \mathcal K-graded vector space \mathcal K {\otimes}_{\mathcal{A}}\mathcal P_{n}. The main goal of the current paper is to explicitly determine an admissible monomial basis of the \mathcal K-graded vector space \mathcal K{\otimes}_{\mathcal{A}}\mathcal P_6 in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree d_{k}=6(2^{k} -1)+9.2^{k} is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, \mbox{Tor}^{\mathcal{A}}_{n, n+d}(\mathcal K, \mathcal K), to the subspace of \mathcal K\otimes_{\mathcal{A}}\mathcal P_{n} consisting of all the GL_n-invariant classes of degree d.


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 Tin, Nguyen Khac