Mathematische Annalen

, Volume 347, Issue 4, pp 765–804 | Cite as

Cohomological stratification of diagram algebras

  • Robert Hartmann
  • Anne Henke
  • Steffen KoenigEmail author
  • Rowena Paget


The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.


Symmetric Group Direct Summand Cell Module Young Module Specht Module 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Brauer R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 854–872 (1937)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cline E., Parshall B., Scott L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cline, E., Parshall, B., Scott, L.: Stratifying endomorphism algebras. Memoir A.M.S. 124 (1996)Google Scholar
  4. 4.
    Diracca L., Koenig S.: Cohomological reduction by split pairs. J. Pure Appl. Algebra 212(3), 471–485 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dlab, V., Ringel, C.M.: The module theoretic approach to quasi-hereditary algebras. In: Representations of Algebras and Related Topics (Kyoto, 1990). London Math. Soc. Lecture Note Ser. 168, pp. 200–224. Cambridge University Press, Cambridge (1992)Google Scholar
  6. 6.
    Doran W.F., Wales D.B.: The partition algebra revisited. J. Algebra 231(1), 265–330 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frisk A.: Dlab’s theorem and tilting modules for stratified algebras. J. Algebra 314(2), 507–537 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Goebel, R., Trlifaj, J.: Approximations and Endomorphism Algebras of Modules. De Gruyter Expositions in Mathematics, vol. 41, xxiv+640 pp. Walter de Gruyter, Berlin (2006)Google Scholar
  9. 9.
    Graham J.J., Lehrer G.I.: Cellular algebras. Invent. Math. 123, 1–34 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Green, J.A.: Polynomial representations of GLn. Lecture Notes in Mathematics, vol. 830. Springer-Verlag, Berlin (1980)Google Scholar
  11. 11.
    Happel D.: Reduction techniques for homological conjectures. Tsukuba J. Math. 17(1), 115–130 (1993)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hartmann R., Paget R.: Young modules and filtration multiplicities for Brauer algebras. Math. Z. 254, 333–357 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hemmer D., Nakano D.: Specht filtrations for Hecke algebras of type A. J. Lond. Math. Soc. 69(2), 623–638 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Henke, A., Paget, R.: Brauer algebras with parameter n = 2 acting on tensor space. Algebr. Represent. Theory 11(6), 545–575 (2008)Google Scholar
  15. 15.
    James, G.D.: Representation Theory of Symmetric Groups. Lecture Notes in Mathematics, vol. 682. Springer, Berlin (1978)Google Scholar
  16. 16.
    Koenig S.: Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J. Pure Appl. Alg. 73, 211–232 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Koenig, S., Xi, C.C.: On the structure of cellular algebras. In: Algebras and Modules II (Geiranger 1996), CMS Conference Proceedings, vol. 24, pp. 365–386. American Mathematical Society (1998)Google Scholar
  18. 18.
    Koenig S., Xi C.C.: Cellular algebras: inflations and Morita equivalences. J. Lond. Math. Soc. 60(2), 700–722 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Koenig S., Xi C.C.: Cellular algebras and quasi-hereditary algebras: a comparison. Elec. Res. Announc. Am. Math. Soc. 5, 71–75 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Koenig S., Xi C.C.: When is a cellular algebra quasi-hereditary. Math. Ann. 315(2), 281–293 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Koenig S., Xi C.C.: A characteristic free approach to Brauer algebras. Trans. Am. Math. Soc. 353, 1489–1505 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Martin P.P.: Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction. J. Knot Theory Ramif. 3(1), 51–82 (1994)zbMATHCrossRefGoogle Scholar
  23. 23.
    Rui H.B.: A criterion on the semisimple Brauer algebras. J. Combin. Theory Ser. A 111(1), 78–88 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Xi C.C.: Partition algebras are cellular. Compos. Math. 119, 99–109 (1999)zbMATHCrossRefGoogle Scholar
  25. 25.
    Xi C.C.: On the quasi-heredity of Birman-Wenzl algebras. Adv. Math. 154(2), 280–298 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Xi, C.C.: On the Finitistic Dimension Conjecture. Advances in Ring Theory, pp. 282–294. World Scientific Publishing, Singapore (2005)Google Scholar
  27. 27.
    Zimmermann-Huisgen B.: Homological domino effects and the first finitistic dimension conjecture. Invent. Math. 108(2), 369–383 (1992)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Robert Hartmann
    • 1
  • Anne Henke
    • 2
  • Steffen Koenig
    • 1
    Email author
  • Rowena Paget
    • 3
  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK
  3. 3.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK

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