Algebras and Representation Theory

, Volume 17, Issue 5, pp 1553–1585 | Cite as

An A-structure on the Cohomology Ring of the Symmetric Group S p with Coefficients in 𝔽 p

  • Stephan Schmid


Let p be a prime. Let 𝔽 p S p be the group algebra of the symmetric group over the finite field with p elements 𝔽 p . Let 𝔽 p be the trivial 𝔽 p S p -module. We choose a projective resolution PRes𝔽 p of the module 𝔽 p and equip the Yoneda algebra \(\mathrm{Ext}^{\ast }_{\mathbb{F}_{p} S_{p}}\left( \mathbb{F}_{p}, \mathbb{F}_{p}\right)\) with an A-structure such that \(\mathrm{Ext}^{\ast }_{\mathbb{F}_{p} S_{p}}\left( \mathbb{F}_{p}, \mathbb{F}_{p}\right)\) becomes a minimal model in the sense of Kadeishvili of the dg-algebra \(\mathrm{Hom}^{\ast }_{\mathbb{F}_{p} S_{p}}\left(PRes \mathbb{F}_{p}, PRes \mathbb{F}_{p}\right)\).


A-infinity Group cohomology Symmetric group Minimal model 

Mathematics Subject Classification (2010)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benson, D.J.: Representations and cohomology I: basic representation theory of finite groups and associative algebras. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  2. 2.
    Bourbaki, N.: Éléments de mathématique - Algèbre - Chapitre 10: Algèbre homologique (1980)Google Scholar
  3. 3.
    Green, J.A.: Walking around the Brauer tree. J. Aust. Math. Soc. 17(2), 197–213 (1974)CrossRefzbMATHGoogle Scholar
  4. 4.
    Gugenheim, V.K.A.M., Lambe, L.A., Stasheff, J.D.: Perturbation theory in differential homological algebra II. Ill. J. Math. 35, 357–373 (1991)MathSciNetzbMATHGoogle Scholar
  5. 5.
    James, G.D.: The representation theory of the symmetric groups. Lecture Notes in Mathematics, vol. 682. Springer-Verlag, Berlin (1978)Google Scholar
  6. 6.
    Johansson, L., Lambe, L.: Transferring algebra structures up to homology equivalence. Math. Scand. 89, 181–200 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kadeishvili, T.V.: On the homology theory of fiber spaces. Russ. Math. Surveys 35(3), 231–238 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kadeishvili, T.V.: Algebraic structure in the homologies of an A(∞)-Algebra (Russian). Soobshch. Akad. Nauk Gruzin. SSR 108(2), 249–252 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kadeishvili, T.V.: The functor D for a category of A(∞)-algebras (Russian). Soobshch. Akad. Nauk Gruzin. SSR 125, 273–276 (1987)MathSciNetGoogle Scholar
  10. 10.
    Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3, 1–35 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Keller, B.: Addendum to ‘Introduction to A-Infinity Algebras and Modules’ (2002)Google Scholar
  12. 12.
    Klamt, A.: A-structures on the Algebra of Extension of Verma Modules in the Parabolic Category 𝒪. Diplomarbeit, Universität Bonn (2010)Google Scholar
  13. 13.
    Künzer, M.: Ties for the Integral Group Ring of the Symmetric Group. Thesis, Bielefeld (1999)Google Scholar
  14. 14.
    Künzer, M.: (Co)homologie von Gruppen, Lecture Notes. Aachen (2006)Google Scholar
  15. 15.
    Lefèvre-Hasegawa, K.: Sur les A-catégories. Thèse de Doctorat, Université Paris VII (2003)Google Scholar
  16. 16.
    Madsen, D.: Homological Aspects in Representation Theory. Thesis, Norwegian University of Science and Technology, Trondheim (2002)Google Scholar
  17. 17.
    Merkulov, S.A.: Strong homotopy algebras of a Kähler manifold. Int. Math. Res. Notices 1999, 153–164 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peel, M.H.: Hook representations of the symmetric groups. Glasgow Math. J. 12, 136–149 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Prouté, A.: Algèbres différentielles fortement homotopiquement associatives. Thèse d’Etat, Université Paris VII (1984)Google Scholar
  20. 20.
    Roggenkamp, K.W.: Integral Representations and Structure of Finite Group Rings. Séminaire de mathématiques supérieures 71. Presses de l’Université de Montréal (1980)Google Scholar
  21. 21.
    Stasheff, J.D.: Homotopy associativity of H-spaces II. Trans. Am. Math. Soc. 108, 293–312 (1963)MathSciNetGoogle Scholar
  22. 22.
    Vejdemo-Johansson, M.: Computation of A-algebras in Group Cohomology. Thesis, Friedrich-Schiller-Universität Jena (2008)Google Scholar
  23. 23.
    Vejdemo-Johansson, M.: A partial A-structure on the cohomology of C m × C n. J. Homotopy Relat. Struct. 3(1), 1–11 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vejdemo-Johansson, M.: Blackbox computation of A-algebras. Georgian Math. J. 17(2), 391–404 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.University of StuttgartStuttgartGermany

Personalised recommendations