Algebras and Representation Theory

, Volume 10, Issue 4, pp 307–314 | Cite as

The Tilting Tensor Product Theorem and Decomposition Numbers for Symmetric Groups



We show how the tilting tensor product theorem for algebraic groups implies a reduction formula for decomposition numbers of the symmetric group. We use this to prove generalisations of various theorems of Erdmann and of James and Williams.


Tilting tensor product theorem Algebraic groups Symmetric group 

Mathematics Subject Classifications (2000)

20C30 20G05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersen, H.H.: Tilting modules for algebraic groups. In: Carter, R., Saxl, J. (eds.) Algebraic Groups and their Representations, vol. 517 of Nato ASI Series, Series C, pp. 25–42 (1998)Google Scholar
  2. 2.
    Andersen, H.H., Jantzen, J.C., Soergel, W.: Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Astérisque 220 (1994)Google Scholar
  3. 3.
    Cox, A.G.: Ext1 for Weyl modules for q-GL(2,k). Math. Proc. Cambridge Philos. Soc. 124, 231–251 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cox, A.G.: Decomposition numbers for distant Weyl modules. J. Algebra 243, 448–472 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dipper, R., James, G.D.: Representations of Hecke algebras of general linear groups. Proc. London Math. Soc. 52(3), 20–52 (1986)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212, 39–60 (1993)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Donkin, S.: The q-Schur algebra, vol. 253 of LMS Lecture Notes Series. Cambridge University Press (1998)Google Scholar
  8. 8.
    Erdmann, K.: Symmetric groups and quasi-hereditary algebras. In: Dlab, V., Scott, L.L. (eds.) Finite Dimensional Algebras and Related Topics, pp. 123–161. Kluwer, Dordrecht (1994)Google Scholar
  9. 9.
    Erdmann, K.: Decomposition numbers for symmetric groups and composition factors of Weyl modules. J. Algebra 180, 316–320 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    James, G.D.: On the decomposition matrices of the symmetric groups, I. J. Algebra 43, 42–44 (1976)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    James, G.D.: Representations of the symmetric group over the field of order 2. J. Algebra 38, 280–308 (1976)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    James, G.D., Kerber, A.: The Representation Theory of the Symmetric Group, vol. 16 of Encyclopedia of Mathematics and its Applications. Addison-Wesley, Reading, MA (1981)Google Scholar
  13. 13.
    James, G.D., Williams, A.L.: Decomposition numbers of symmetric groups by induction. J. Algebra 228, 119–142 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jantzen, J.C.: Representations of algebraic groups, vol. 107 of Mathematical Surveys and Monographs. AMS, 2nd edn. (2003) (note: all citations except those involving tilting modules can also be found in the first edition)Google Scholar
  15. 15.
    Jensen, J.G.: On the character of some modular indecomposable tilting modules for SL3. J. Algebra 232, 397–419 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Comm. Math. Phys. 181, 205–263 (1996)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Leclerc, B.: Decomposition numbers and canonical bases. Algebr. Represent. Theory 3, 277–287 (2000)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lusztig, G.: Some problems in the representation theory of finite Chevalley groups. In: Mason, G., Cooperstein, B. (eds.) The Santa Cruz Conference on Finite Groups, vol. 37, pp. 313–317 (1979)Google Scholar
  19. 19.
    Rasmussen, T.E.: Multiplicities of second cell tilting modules. J. Algebra 288, 1–19 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–225 (1991)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren. Represent. Theory 1, 115–132 (1997)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric for tilting modules. Represent. Theory 1, 83–114 (1997)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1957)MathSciNetGoogle Scholar
  24. 24.
    To Law, K.W.: Results on decomposition matrices for the symmetric groups. Ph.D. thesis, Cambridge (1983)Google Scholar
  25. 25.
    Williams, A.L.: Symmetric group decomposition numbers for some three-part partitions. Comm. Algebra 34, 1599–1613 (2006)MATHMathSciNetGoogle Scholar
  26. 26.
    Xanthopoulos, S.: On a question of Verma about indecomposable representations of algebraic groups and of their Lie algebras. Ph.D. thesis, London (1992)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Centre for Mathematical ScienceCity UniversityLondonEngland

Personalised recommendations