Skip to main content

Branching Rules for Symmetric Groups and Applications

  • Chapter
Algebraic Groups and their Representations

Part of the book series: NATO ASI Series ((ASIC,volume 517))

Abstract

Let F be an arbitrary field of characteristic p ≥ 0, and Σ n be the symmetric group on n letters. We shall discuss finite-dimensional representations of Σ n over F or, equivalently, finite-dimensional FΣ n -modules. Basic facts important for our topic can be found in [18, 21, 20]. We recall here some of them.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amitsur S.A. (1979) The polynomial identities of associative rings, in A. Goldie and T. Willmore (eds.), Noetherian Rings and Rings with Polynomial Identities, Proc. Conf. Univ. Durham, pp. 1–38.

    Google Scholar 

  2. Baranov, A. and Kleshchev, A. (1997) Maximal ideals in modular group algebras of the finitary symmetric and alternating groups, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  3. Benson, D. (1987) Some remarks on the decomposition numbers for the symmetric groups, in P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47 (1), Amer. Math. Soc., Providence, pp. 381–394.

    Google Scholar 

  4. Bessenrodt, C. and Olsson, J.B. (1996) On residue symbols and the Mullineux conjecture, preprint, University of Copenhagen (to appear in J. Alg. Comb.).

    Google Scholar 

  5. Brundan, J. (1997) Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London Math. Soc., to appear.

    Google Scholar 

  6. Brundan, J., Kleshchev, A. and Suprunenko, I. (1997) Semisimple restrictions from GL(n) to GL(n - 1), preprint.

    Google Scholar 

  7. Dipper, R. (1990) On quotients of Hom-functors and representations of finite general linear groups, I, J. Algebra 130, 235–259.

    Article  MathSciNet  MATH  Google Scholar 

  8. Erdmann, K. (1995) Tensor products and dimensions of simple modules for symmetric groups, Manuscripta Math. 88, 357–386.

    Article  MathSciNet  MATH  Google Scholar 

  9. Erdmann, K. (1996) Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180, 316–320.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ford, B. (1995) Irreducible restrictions of representations of the symmetric groups, Bull. London Math. Soc. 27, 453–459.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ford, B. (1997) Irreducible representations of the alternating groups in odd characteristic, Proc. Amer. Math. Soc. 125, 375–380.

    MATH  Google Scholar 

  12. Ford, B. and Kleshchev, A. (1997) A proof of the Mullineux conjecture, Math. Z. 226, 267–308.

    Article  MathSciNet  MATH  Google Scholar 

  13. Formanek, E. and Lawrence, D. (1976) The group algebra of the infinite symmetric group, Israel J. Math. 23, 325–331.

    Article  MathSciNet  MATH  Google Scholar 

  14. Formanek, E. and. Procesi, C. (1976) Mumford’s conjecture for the general linear group, Adv. in Math. 19, 292–305.

    Article  MathSciNet  MATH  Google Scholar 

  15. Green, J.A. (1980) Polynomial Representations of GL n , Lecture Notes in Mathematics 830, Springer-Verlag, Berlin, Heidelberg, New York.

    MATH  Google Scholar 

  16. James, G.D. (1976) On the decomposition matrices of the symmetric groups, II, J. Algebra 43, 45–54.

    Article  MathSciNet  MATH  Google Scholar 

  17. James, G.D. (1978) On a conjecture of Carter concerning irreducible Specht modules, Math. Proc. Carob. Phil. Soc. 83, 11–17.

    Article  MATH  Google Scholar 

  18. James, G.D. (1978) The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, Heidelberg, New York.

    MATH  Google Scholar 

  19. James, G.D. (1983) On the minimal dimensions of irreducible representations of symmetric groups. Math. Proc. Cambridge Philos. Soc. 94, 417–424.

    Article  MathSciNet  MATH  Google Scholar 

  20. James, G.D. (1987) The representation theory of the symmetric groups, in P. Fong (ed.), The Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47 (1), Amer. Math. Soc., Providence, pp. 111–126.

    Google Scholar 

  21. James, G.D. and Kerber, A. (1981) The Representation Theory of the Symmetric Group, Addison-Wesley, London.

    MATH  Google Scholar 

  22. James, G.D. and Murphy, G.E. (1979) The determinant of the Gram matrix for a Specht module. J. Algebra 59, 222–235.

    Article  MathSciNet  MATH  Google Scholar 

  23. Jantzen, J.C. and Seitz, G.M. (1992) On the representation theory of the symmetric groups, Proc. London Math. Soc. 65, 475–504.

    Article  MathSciNet  MATH  Google Scholar 

  24. Kemer, A. (1996) Remarks on the prime varieties, Israel J. Math. 96, 341–356.

    Article  MathSciNet  MATH  Google Scholar 

  25. Kleshchev, A.S. (1994) On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I, Proc. London Math. Soc. 69, 515–540.

    Article  MathSciNet  MATH  Google Scholar 

  26. Kleshchev, A.S. (1994) On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II, Comm. Alg. 22, 6175–6208.

    Article  MathSciNet  MATH  Google Scholar 

  27. Kleshchev, A.S. (1995) Branching rules for modular representations of symmetric groups, I, J. Algebra 178, 493–511.

    Article  MathSciNet  MATH  Google Scholar 

  28. Kleshchev, A.S. (1995) Branching rules for modular representations of symmetric groups, II, J. Reine Angew. Math. 459, 163–212.

    MathSciNet  MATH  Google Scholar 

  29. Kleshchev, A.S. (1996) Branching rules for modular representations of symmetric groups, III, J. London Math. Soc. 54, 25–38.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kleshchev, A.S. (1998) Branching rules for modular representations of symmetric groups, IV, J. Algebra 201, 547–572.

    Article  MathSciNet  MATH  Google Scholar 

  31. Kleshchev, A.S. (1996) Completely splittable representations of symmetric groups, J. Algebra 181, 584–592.

    Article  MathSciNet  MATH  Google Scholar 

  32. Kleshchev, A.S. (1997) On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. Lond. Math. Soc. 75, 497–558.

    Article  MathSciNet  MATH  Google Scholar 

  33. Kleshchev, A. and Premet, A. (1997) The globally irreducible representations of symmetric groups, Math. Proc. Camb. Phil. Soc., to appear.

    Google Scholar 

  34. Lascoux, A., Leclerc, B. and Thibon, J.-Y. (1996) Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181, 205–263.

    Article  MathSciNet  MATH  Google Scholar 

  35. Mathieu, O. (1996) On the dimension of some modular irreducible representations of the symmetric group, Lett. Math. Phys. 38, 23–32.

    Article  MathSciNet  MATH  Google Scholar 

  36. Mathieu, O. and Papadopoulo, G. (1997) A character formula for a family of simple modular representations of GL n , preprint.

    Google Scholar 

  37. Mullineux, G. (1979) Bijections of p-regular partitions and p-modular irreducibles of the symmetric groups, J. London Math. Soc. 20, 60–66.

    Article  MathSciNet  MATH  Google Scholar 

  38. Mullineux, G. (1979) On the p-cores of p-regular diagrams, J. London Math. Soc. 20, 222–226.

    Article  MathSciNet  MATH  Google Scholar 

  39. Razmyslov, Yu.P. (1974) Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR. Izvestiya 8, 727–760.

    Article  Google Scholar 

  40. Razmyslov, Yu.P. (1989) Identities of Algebras and Their Representations, Nauka, Moscow (in Russian).

    MATH  Google Scholar 

  41. Ryba, A. (1994) Fibonacci representations of the symmetric groups, J. Algebra 170, 678–686.

    Article  MathSciNet  MATH  Google Scholar 

  42. Seitz, G.M. (1987) The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 365, 1–286.

    MathSciNet  Google Scholar 

  43. Seitz, G.M. (1992) Subgroups of finite and algebraic groups, in Groups, Combinatorics, and Geometry, London Math. Soc. Lecture Notes Series 165, Cambridge University Press, pp. 316–326.

    Chapter  Google Scholar 

  44. Sheth, J. (1997) Branching rules for two-part partitions and inductive systems, preprint, University of Oregon.

    Google Scholar 

  45. Smith, S.D. (1982) Irreducible modules and parabolic subgroups. J. Algebra 74, 286–289.

    Article  Google Scholar 

  46. Xu, M. (1997) On Mullineux’ conjecture in the representation theory of symmetric groups, Comm. Alg. 25, 1797–1803.

    Article  MATH  Google Scholar 

  47. Zalesskiĭ, A.E. (1995) Group rings of simple locally finite groups, in B. Hartley et al. (eds.) Finite and Locally Finite Groups, Kluwer Academic Publishers, Dordrecht, pp. 219–246.

    Chapter  Google Scholar 

  48. Zalesskiĭ, A.E. (1996) Modular group rings of the finitary symmetric groups, Israel J. Math. 96, 609–621.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Kleshchev, A.S. (1998). Branching Rules for Symmetric Groups and Applications. In: Carter, R.W., Saxl, J. (eds) Algebraic Groups and their Representations. NATO ASI Series, vol 517. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5308-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5308-9_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-5292-1

  • Online ISBN: 978-94-011-5308-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics