Skip to main content
SpringerLink
Log in

The complexity of the Specht modules corresponding to hook partitions

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E s and of Specht modules corresponding to hook partitions μ restricted to E s where s is the p-weight of μ.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alperin J.L., Evens L.: Representations, resolutions, and Quillen’s dimension theorem. J. Pure Appl. Algebra 22, 1–9 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Avrunin G.S., Scott L.L.: Quillen stractification for modules. Invent. Math. 66, 277–286 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. D. J. Benson, Representations and Cohomology I, II, Cambridge Studies in Advanced Mathematics, 30, 31, Cambridge Univ. Press 1998.

  4. J. F. Carlson, The complexity and varieties of modules, Integral representations and their applications, Oberwolfach 1980. Lecture Notes in Math., 882, Springer-Verlag, 1981, 415–422.

  5. Carlson J.F.: The varieties and the cohomology ring of a module. J. Algebra 85, 104–143 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Donkin, Symmetric and exterior powers, linear source modules and representations of Schur superalgebras, Proc. London Math. Soc. (3) 83 (2001) 647–680.

  7. L. Evens, The Cohomology of Groups, Oxford Science Publications, 1991.

  8. Friedlander E.M., Pevtsova J., Suslin A.: Generic and maximal Jordan types. Invent. Math. 168, 485–522 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., 682, Springer-Verlag, 1978.

  10. G. D. James and A. Kerber, The Representation Theory of the Symmetric Groups, Encyclopedia of Math. App., 16, Addison-Wesley Publishing Company, 1981.

  11. D. J. Hemmer, The complexity of certain Specht modules for the symmetric group, J. Algebraic Combin., in press.

  12. Hemmer D.J., Nakano D.K.: Support varieties for modules over symmetric groups. J. Algebra 254, 422–440 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lim K.J.: The varieties for some Specht modules. J. Algebra 321, 2287–2301 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wheeler W.W.: Generic Module Theory. J. Algebra 185, 205–228 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wildon M.: Two theorems on the vertices of Specht modules. Arch. Math. 81, 505–511 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Kay Jin Lim

Authors
  1. Kay Jin Lim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kay Jin Lim.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lim, K.J. The complexity of the Specht modules corresponding to hook partitions. Arch. Math. 93, 11–22 (2009). https://doi.org/10.1007/s00013-009-0011-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-009-0011-x

Mathematics Subject Classification (2000)

Keywords

Search

Navigation