The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

Nguyễn H.V. Hưng

doi:10.1016/j.crma.2010.07.032

Topology

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra
[Homorphismes entre l'algèbre de Dickson–Mùi comme module sur l'algèbre de Steenrod]

Présenté par : Christophe Soulé

Nguyễn H.V. Hưng ¹

¹ Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Viet Nam

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1001-1004.

Résumé
Abstract

L'algèbre de Dickson–Mùi consiste en les invariants sous l'action du groupe linéaire dans l'algèbre de cohomologie modulo p d'un p-groupe abélien élémentaire. C'est un module sur l'algèbre de Steenrod $A$ . Nous déterminons explicitement tous les homorphismes $A$ -linéaires entre ces algèbres ainsi que leurs automorphismes ( $A$ -linéaires).

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, $A$ . We determine explicitly all the $A$ -module homomorphisms between the Dickson–Mùi algebras and all the $A$ -module automorphisms of these algebras.

Reçu le : 2010-04-27
Accepté le : 2010-07-30
Publié le : 2010-09-01

DOI : 10.1016/j.crma.2010.07.032

Affiliations des auteurs :

Nguyễn H.V. Hưng ¹

¹ Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyễn Trãi Street, Hanoi, Viet Nam

@article{CRMATH_2010__348_17-18_1001_0,
     author = {Nguyễn H.V. Hưng},
     title = {The homomorphisms between the {Dickson{\textendash}M\`ui} algebras as modules over the {Steenrod} algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1001--1004},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.032},
     language = {en},
}

TY  - JOUR
AU  - Nguyễn H.V. Hưng
TI  - The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1001
EP  - 1004
VL  - 348
IS  - 17-18
PB  - Elsevier
DO  - 10.1016/j.crma.2010.07.032
LA  - en
ID  - CRMATH_2010__348_17-18_1001_0
ER  -

%0 Journal Article
%A Nguyễn H.V. Hưng
%T The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra
%J Comptes Rendus. Mathématique
%D 2010
%P 1001-1004
%V 348
%N 17-18
%I Elsevier
%R 10.1016/j.crma.2010.07.032
%G en
%F CRMATH_2010__348_17-18_1001_0

Nguyễn H.V. Hưng. The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1001-1004. doi : 10.1016/j.crma.2010.07.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.032/

Bibliographie
Cité par

[1] J.F. Adams; J.H. Gunawardena; H.R. Miller The Segal conjecture for elementary abelian p-groups, Topology, Volume 24 (1985), pp. 435-460 (MR0816524)

[2] G. Carlsson G.B. Segal's Burnside ring conjecture for ${(Z / 2)}^{k}$ , Topology, Volume 22 (1983), pp. 83-103 (MR0682060)

[3] 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., Volume 12 (1911), pp. 75-98 (MR1500882)

[4] Nguyễn H.V. Hưng The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc., Volume 113 (1991), pp. 1097-1104 (MR1064904)

[5] Nguyễn H.V. Hưng; Pham A. Minh The action of the mod p Steenrod operations on the modular invariants of linear groups, Vietnam J. Math., Volume 23 (1995), pp. 39-56 (MR1367491)

[6] Nguyễn H.V. Hưng; F.P. Peterson Spherical classes and the Dickson algebra, Math. Proc. Cambridge Philos. Soc., Volume 124 (1998), pp. 253-264 (MR1631123)

[7] N.E. Kechagias A Steenrod–Milnor action ordering on Dickson invariants www.math.uoi.gr/~nondas_k (manuscript posted on his webpage)

[8] H.R. Miller The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2), Volume 120 (1984), pp. 39-87 (MR0750716)

[9] Huỳnh Mùi Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 22 (1975), pp. 319-369 (MR0422451)

Cité par Sources :

^☆ The work was supported in part by a grant of the NAFOSTED.

Commentaires - Politique