Startseite Symmetric powers of Nat SL(2,𝕂)
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Symmetric powers of Nat SL(2,𝕂)

  • Adrien Deloro EMAIL logo
VerĂśffentlicht/Copyright: 14. Januar 2016
Journal of Group Theory
Aus der Zeitschrift Journal of Group Theory Band 19 Heft 4

Abstract

In this paper, we identify the representations 𝕂⁢[Xk,Xk-1⁢Y,…,Yk] among abstract ℤ⁢[SL2⁡(𝕂)]-modules. One result is on ℚ⁢[SL2⁡(ℤ)]-modules of nilpotence length at most 5 and generalises a classical “quadratic” theorem by Smith and Timmesfeld; there are reasons to believe that this generalisation is close to optimal. Another result is on extending the linear structure on the module from the prime field to 𝕂. All proofs are by computation in the group ring using the Steinberg relations.

References

[1] Andersen H. H., Jørgensen J. and Landrock P., The projective indecomposable modules of SL⁢(2,pn), Proc. Lond. Math. Soc. (3) 46 (1983), no. 1, 38–52. 10.1112/plms/s3-46.1.38Suche in Google Scholar

[2] Cherlin G. and Deloro A., Small representations of SL2 in the finite Morley rank category, J. Symb. Log. 77 (2012), no. 3, 919–933. 10.2178/jsl/1344862167Suche in Google Scholar

[3] Deloro A., Changes to a theorem of Timmesfeld. I: Quadratic actions, Confluentes Math. 5 (2013), no. 2, 23–41. Suche in Google Scholar

[4] Deloro A., Symmetric powers of Nat⁡𝔰⁢𝔩2⁢(𝕂), Comm. Algebra (2016), 10.1080/ 00927872.2014.900690. 10.1080/ 00927872.2014.900690Suche in Google Scholar

[5] GrĂźninger M., On cubic action of a rank one group, preprint 2011, http://arxiv.org/abs/1106.2310v2. 10.1090/memo/1356Suche in Google Scholar

[6] Lyndon R. C. and Schupp P. E., Combinatorial Group Theory, Classics Math., Springer, Berlin, 2001. 10.1007/978-3-642-61896-3Suche in Google Scholar

[7] Serre J.-P., Trees, Springer Monogr. Math., Springer, Berlin, 2003. Suche in Google Scholar

[8] Smith S. D., Quadratic action and the natural module for SL2⁢(k), J. Algebra 127 (1989), no. 1, 155–162. 10.1016/0021-8693(89)90280-9Suche in Google Scholar

[9] Timmesfeld F. G., Abstract Root Subgroups and Simple Groups of Lie Type, Monogr. Math. 95, Birkhäuser, Basel, 2001. 10.1007/978-3-0348-7594-3Suche in Google Scholar

Received: 2015-4-30
Revised: 2015-10-1
Published Online: 2016-1-14
Published in Print: 2016-7-1

Š 2016 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Codegrees and nilpotence class of p-groups
  3. On finite 2-equilibrated p-groups
  4. Minimal dimension of faithful representations for p-groups
  5. On the Christensen–Wang bounds for the ghost number of a p-group algebra
  6. A family of non-Schurian p-Schur rings over groups of order p3
  7. Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup
  8. Symmetric powers of Nat SL(2,𝕂)
  9. Nested GVZ-groups
  10. On some groups generated by involutions
  11. Thompson’s conjecture for finite simple groups of Lie type Bn and Cn
https://doi.org/10.1515/jgth-2015-0049

Artikel in diesem Heft

  1. Frontmatter
  2. Codegrees and nilpotence class of p-groups
  3. On finite 2-equilibrated p-groups
  4. Minimal dimension of faithful representations for p-groups
  5. On the Christensen–Wang bounds for the ghost number of a p-group algebra
  6. A family of non-Schurian p-Schur rings over groups of order p3
  7. Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup
  8. Symmetric powers of Nat SL(2,𝕂)
  9. Nested GVZ-groups
  10. On some groups generated by involutions
  11. Thompson’s conjecture for finite simple groups of Lie type Bn and Cn
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern kĂśnnen?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 21.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jgth-2015-0049/html
Button zum nach oben scrollen