The mod $p$ Margolis homology of the Dickson–Mùi algebra

Nguyễn H. V. Hưng

doi:10.5802/crmath.151

Algèbre homologique

The mod

p

Margolis homology of the Dickson–Mùi algebra

Nguyễn H. V. Hưng ¹

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

Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 229-236.

Résumé
Abstract

Soit $E^{n} = {(ℤ / p)}^{n}$ le groupe agissant sur lui même par les translations. On le considère comme sous-groupe du groupe symétrique $𝕊_{p^{n}}$ en $p^{n}$ lettres. Dans cette note on calcule entièrement l’homologie de Margolis modulo $p$ de l’algèbre de Dickson–Mùi, i.e. l’homologie de l’image de la restriction $R e s (𝕊_{p^{n}}, E^{n}) : H^{*} (𝕊_{p^{n}}; 𝔽_{p}) \to H^{*} (E^{n}; 𝔽_{p})$ en choisissant pour différentielles les opérations de Milnor $Q_{j}$ , pour $p$ un nombre premier impair et pour tout $n$ , $j$ . La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava $K {(j)}^{*} (B 𝕊_{m})$ du groupe symétrique $𝕊_{m}$ en $m$ lettres. L’outil principal de notre travail est la notion d’éléments « critiques ». L’homologie de Margolis mod $p$ de l’algèbre de Dickson–Mùi concentre en degrés pairs. Elle est analogue à l’homologie de Margolis mod $2$ de l’algèbre de Dickson.

Let $E^{n} = {(ℤ / p)}^{n}$ be regarded as the translation group on itself. It is considered as a subgroup of the symmetric group $𝕊_{p^{n}}$ on $p^{n}$ letters. We completely compute the mod $p$ Margolis homology of the Dickson–Mùi algebra, i.e. the homology of the image of the restriction $R e s (𝕊_{p^{n}}, E^{n}) : H^{*} (𝕊_{p^{n}}; 𝔽_{p}) \to H^{*} (E^{n}; 𝔽_{p})$ with the differential to be the Milnor operation $Q_{j}$ , for $p$ an odd prime and for any $n$ , $j$ . The motivation for this problem is that, the Margolis homology of the Dickson–Mùi algebra plays a key role in study of the Morava K-theory $K {(j)}^{*} (B 𝕊_{m})$ of the symmetric group $𝕊_{m}$ on $m$ letters. The main tool of our work is the notion of “critical” elements. The mod $p$ Margolis homology of the Dickson–Mùi algebra concentrates on even degrees. It is analogous to the mod $2$ Margolis homology of the Dickson algebra.

Reçu le : 2020-07-22
Révisé le : 2020-10-05
Accepté le : 2020-11-15
Publié le : 2021-04-20

DOI : 10.5802/crmath.151

Classification : 55S05, 55S10, 55N99

Affiliations des auteurs :

Nguyễn H. V. Hưng ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Nguyễn H. V. Hưng},
     title = {The mod $p$ {Margolis} homology of the {Dickson{\textendash}M\`ui} algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--236},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.151},
     language = {en},
}

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Nguyễn H. V. Hưng. The mod $p$ Margolis homology of the Dickson–Mùi algebra. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 229-236. doi : 10.5802/crmath.151. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.151/

Bibliographie
Cité par

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[2] N. H. V. Hưng The Margolis homology of the Dickson algebra and Pengelley–Sinha’s Conjecture (submitted)

[3] N. H. V. Hưng The action of the Steenrod squares on the modular invariants of linear groups, Proc. Am. Math. Soc., Volume 113 (1991) no. 4, pp. 1097-1104 | DOI | MR | Zbl

[4] N. H. V. Hưng The mod 2 Margolis homology of the Dickson algebra, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 4, pp. 505-510 | MR | Zbl

[5] N. H. V. Hưng; Phạm Anh Minh The action of the mod p Steenrod operations on the modular invariants of linear groups, Vietnam J. Math., Volume 23 (1995) no. 1, pp. 39-56 | MR | Zbl

[6] John W. Milnor The Steenrod algebra and its dual, Ann. Math., Volume 67 (1958), pp. 150-171 | DOI | MR | Zbl

[7] Huỳnh Mùi Modular invariant theory and the cohomology algebras of symmetric group, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 22 (1975), pp. 319-369 | MR | Zbl

[8] Dev P. Sinha Cohomology of symmetric groups (2019) (Lecture on the Vietnam-US Mathematical joint Metting, Quynhon June 10-13)

[9] Nobuaki Yagita On the Steenrod algebra of Morava K-theory, J. Lond. Math. Soc., Volume 22 (1980), pp. 423-438 | DOI | MR | Zbl

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