previous :: next
The squaring operation on ${\cal A}$-generators of the Dickson algebra
Nguyên H. V. Hưng and Võ T. N. Quỳnh
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 85, Number 6
(2009), 67-70.
Abstract
We study the squaring operation $Sq^0$ on the dual of the minimal ${\cal A}$-generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes-Zarati homomorphism vanishes (1) on every element in any finite $Sq^0$-family in $Ext_{\cal A}^*({\bf F}_2, {\bf F}_2)$ except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.
First Page:
Show
Hide
Keywords: Modular representations; invariant theory; cohomology of the Steenrod algebra; spherical classes; Lannes-Zarati homomorphism
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.pja/1244037799
Digital Object Identifier: doi:10.3792/pjaa.85.67
Mathematical Reviews number (MathSciNet): MR2532421
References
J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20--104.
Mathematical Reviews (MathSciNet): MR141119
Digital Object Identifier: doi:10.2307/1970147
JSTOR: links.jstor.org
J. M. Boardman, Modular representations on the homology of powers of real projective space, in Algebraic topology (Oaxtepec, 1991), Contemp. Math. 146, (1993), Amer. Math. Soc., Providence, RI, pp. 49--70.
Mathematical Reviews (MathSciNet): MR1224907
Zentralblatt MATH: 0789.55015
W. Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157--186.
Mathematical Reviews (MathSciNet): MR251736
Digital Object Identifier: doi:10.2307/1970686
JSTOR: links.jstor.org
R. R. Bruner, The cohomology of the mod 2 Steenrod algebra: A computer calculation, WSU Research Report 37 (1997), pp. 217.
R. R. Bruner, L. M. Hà and N. H. V. H\uw ng, On the behavior of the algebraic transfer, Trans. Amer. Math. Soc. 357 (2005), no. 2, 473--487. (electronic).
Mathematical Reviews (MathSciNet): MR2095619
Digital Object Identifier: doi:10.1090/S0002-9947-04-03661-X
Zentralblatt MATH: 1055.55015
E. B. Curtis, The Dyer-Lashof algebra and the $\Lambda $-algebra, Illinois J. Math. 19 (1975), 231--246.
Mathematical Reviews (MathSciNet): MR377885
Zentralblatt MATH: 0311.55007
L. E. Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc. 12 (1911), 75--98.
Mathematical Reviews (MathSciNet): MR1500882
Digital Object Identifier: doi:10.2307/1988736
JSTOR: links.jstor.org
V. Giambalvo and F. P. Peterson, $\cala$-generators for ideals in the Dickson algebra, J. Pure Appl. Algebra 158 (2001), no. 2--3, 161--182.
Mathematical Reviews (MathSciNet): MR1822839
Digital Object Identifier: doi:10.1016/S0022-4049(00)00051-7
Zentralblatt MATH: 0984.55012
P. G. Goerss, Unstable projectives and stable $\operatornameExt:$ with applications, Proc. London Math. Soc. (3) 53 (1986), no. 3, 539--561.
Mathematical Reviews (MathSciNet): MR868458
Digital Object Identifier: doi:10.1112/plms/s3-53.3.539
Zentralblatt MATH: 0638.55018
L. M. Hà, Sub-Hopf algebras of the Steenrod algebra and the Singer transfer, (Proceedings of the School and Conference in Algebraic Topology}, 11 (2007), Geom. Topol. Publ., Coventry, pp. 81--105.
Mathematical Reviews (MathSciNet): MR2402802
N. H. V. H\uw ng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3893--3910.
Mathematical Reviews (MathSciNet): MR1433119
Digital Object Identifier: doi:10.1090/S0002-9947-97-01991-0
JSTOR: links.jstor.org
Zentralblatt MATH: 0902.55004
N. H. V. H\uw ng, On triviality of Dickson invariants in the homology of the Steenrod algebra, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 1, 103--113.
Mathematical Reviews (MathSciNet): MR1937796
Zentralblatt MATH: 1040.55004
N. H. V. H\uw ng, The cohomology of the Steenrod algebra and representations of the general linear groups, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4065--4089. (electronic).
Mathematical Reviews (MathSciNet): MR2159700
Digital Object Identifier: doi:10.1090/S0002-9947-05-03889-4
Zentralblatt MATH: 1074.55006
N. H. V. H\uw ng and Tråh n N. Nam, The hit problem for the Dickson algebra, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5029--5040. (electronic).
Mathematical Reviews (MathSciNet): MR1852092
Digital Object Identifier: doi:10.1090/S0002-9947-01-02705-2
JSTOR: links.jstor.org
Zentralblatt MATH: 0979.55011
N. H. V. H\uw ng and F. P. Peterson, $\cala$--generators for the Dickson algebra, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4687--4728.
Mathematical Reviews (MathSciNet): MR1316852
Digital Object Identifier: doi:10.2307/2155059
JSTOR: links.jstor.org
Zentralblatt MATH: 0864.55016
N. H. V. H\uw ng and F. P. Peterson, Spherical classes and the Dickson algebra, Math. Proc. Camb. Philos. Soc. 124 (1998), no. 2, 253--264.
Mathematical Reviews (MathSciNet): MR1631123
Digital Object Identifier: doi:10.1017/S0305004198002667
Zentralblatt MATH: 0906.55013
N. H. V. H\uw ng and Võ T. N. Qu\`ynh, The image of the fourth algebraic transfer. (Submitted)
J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation}, Math. Zeit. 194 (1987), no. 1, 25--59.
Mathematical Reviews (MathSciNet): MR871217
Digital Object Identifier: doi:10.1007/BF01168004
W.-H. Lin and M. Mahowald, The Adams spectral sequence for Minami's theorem, in Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), Contemp. Math., 220 (1998), Amer. Math. Soc., Providence, RI. pp. 143--177.
Mathematical Reviews (MathSciNet): MR1642893
Zentralblatt MATH: 0916.55011
T. N. Nam, Transfert algébrique et représentation modulaire du groupe linéare, Ann. Inst. Fourier, 58 (2008), 1785--1837.
Mathematical Reviews (MathSciNet): MR2445834
W. M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493--523.
Mathematical Reviews (MathSciNet): MR1022818
Digital Object Identifier: doi:10.1007/BF01221587
Zentralblatt MATH: 0687.55014
M. C. Tangora, On the cohomology of the Steenrod algebra, Math. Z. 116 (1970), 18--64.
Mathematical Reviews (MathSciNet): MR266205
Digital Object Identifier: doi:10.1007/BF01110185
Zentralblatt MATH: 0198.28202
J. S. P. Wang, On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one, Illinois J. Math. 11 (1967), 480--490.
Mathematical Reviews (MathSciNet): MR214065
Zentralblatt MATH: 0161.20204
previous :: next
Proceedings of the Japan Academy, Series A, Mathematical Sciences
