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On a Minimal Set of Generators for the Polynomial Algebra of Five Variables as a Module over the Steenrod Algebra

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Abstract

Let P k be the graded polynomial algebra \(\mathbb {F}_{2}[x_{1},x_{2},{\ldots } ,x_{k}]\) over the prime field of two elements, \(\mathbb {F}_{2}\), with the degree of each x i being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for P k as a module over the mod-2 Steenrod algebra, \(\mathcal {A}\). In this paper, we explicitly determine a minimal set of \(\mathcal {A}\)-generators for P k in the case k = 5 and the degree 4(2d−1) with d an arbitrary positive integer.

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References

  1. Boardman, J.M.: Modular representations on the homology of power of real projective space. In: Tangora, M.C. (ed.) Algebraic Topology, Oaxtepec, 1991. Contemp. Math., vol. 146, pp. 49–70 (1993)

  2. Bruner, R.R., Hà, L.M., Hu’ng, N.H.V.: On behavior of the algebraic transfer. Trans. Am. Math. Soc. 357, 473–487 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carlisle, D.P., Wood, R.M.W.: The boundedness conjecture for the action of the Steenrod algebra on polynomials. In: Ray, N., Walker, G. (eds.) Adams Memorial Symposium on Algebraic Topology 2, (Manchester, 1990). London Math. Soc. Lecture Notes Ser., vol. 176, pp. 203–216. Cambridge Univ. Press, Cambridge (1992)

  4. Crabb, M.C., Hubbuck, J.R.: Representations of the homology of BV and the Steenrod algebra II. Algebraic Topology: new trend in localization and periodicity. Progr. Math. 136, 143–154 (1996)

    MATH  Google Scholar 

  5. Hu’ng, N.H.V.: The cohomology of the Steenrod algebra and representations of the general linear groups. Trans. Am. Math. Soc. 357, 4065–4089 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu’ng, N.H.V., Nam, T.N.: The hit problem for the modular invariants of linear groups. J. Algebra 246, 367–384 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Janfada, A.S., Wood, R.M.W.: The hit problem for symmetric polynomials over the Steenrod algebra. Math. Proc. Camb. Philos. Soc. 133, 295–303 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kameko, M.: Products of projective spaces as Steenrod modules. PhD Thesis, The Johns Hopkins University (1990)

  9. Mothebe, M.F.: Generators of the polynomial algebra \(\mathbb {F}_{2}[x_{1},\ldots , x_{n}]\) as a module over the Steenrod algebra. PhD Thesis, The University of Manchester (1997)

  10. Nam, T.N.: \(\mathcal {A}\)-générateurs génériquess pour l’algèbre polynomiale. Adv. Math. 186, 334–362 (2004)

  11. Peterson, F.P.: Generators of \(H^{*}(\mathbb RP^{\infty } \times \mathbb RP^{\infty })\) as a module over the Steenrod algebra. Abstr. Am. Math. Soc. 833, 55–89 (1987)

  12. Phuc, D.V., Sum, N.: On the generators of the polynomial algebra as a module over the Steenrod algebra. C. R. Acad. Sci. Paris, Ser. I 353, 1035–1040 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Phuc, D.V., Sum, N.: On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra. Quy Nhon University, Viet Nam, preprint. arXiv:1502.05569 (2013)

  14. Priddy, S.: On characterizing summands in the classifying space of a group. I. Am. J. Math. 112, 737–748 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Repka, J., Selick, P.: On the subalgebra of \(H_{*}((\mathbb RP^{\infty })^{n};\mathbb {F}_{2})\) annihilated by Steenrod operations. J. Pure Appl. Algebra 127, 273–288 (1998)

  16. Silverman, J.H.: Hit polynomials and the canonical antiautomorphism of the Steenrod algebra. Proc. Am. Math. Soc. 123, 627–637 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Silverman, J.H., Singer, W.M.: On the action of Steenrod squares on polynomial algebras II. J. Pure Appl. Algebra 98, 95–103 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Singer, W.M.: The transfer in homological algebra. Math. Z. 202, 493–523 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Singer, W.M.: On the action of the Steenrod squares on polynomial algebras. Proc. Am. Math. Soc. 111, 577–583 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Steenrod, N.E., Epstein, D.B.A.: Cohomology operations, annals of mathematics studies, vol. 50. Princeton University Press, Princeton N.J (1962)

    Google Scholar 

  21. Sum, N.: The negative answer to Kameko’s conjecture on the hit problem. Adv. Math. 225, 2365–2390 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sum, N.: On the Peterson hit problem. Adv. Math. 274, 432–489 (2015)

    Article  MathSciNet  Google Scholar 

  23. Walker, G., Wood, R.M.W.: Young tableaux and the Steenrod algebra. In: Proceedings of the International School and Conference in Algebraic Topology, Ha Noi 2004, Geom. Topol. Monogr., Math. Sci. Publ., Coventry, vol. 11, pp 379–397 (2007)

  24. Walker, G., Wood, R.M.W.: Flag modules and the hit problem for the Steenrod algebra. Math. Proc. Camb. Philos. Soc. 147, 143–171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wood, R.M.W.: Steenrod squares of polynomials and the Peterson conjecture. Math. Proc. Camb. Philos. Soc. 105, 307–309 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wood, R.M.W.: Problems in the Steenrod algebra. Bull. London Math. Soc. 30, 449–517 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The final version of this paper was completed while the second named author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM) from August to December 2015. He would like to thank the VIASM for financial support and kind hospitality. The work was supported in part by a grant of the NAFOSTED.

We would like to express our warmest thanks to the referee for the careful reading and helpful suggestions.

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Phuc, D.V., Sum, N. On a Minimal Set of Generators for the Polynomial Algebra of Five Variables as a Module over the Steenrod Algebra. Acta Math Vietnam 42, 149–162 (2017). https://doi.org/10.1007/s40306-016-0190-z

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  • DOI: https://doi.org/10.1007/s40306-016-0190-z

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