Skip to main content
SpringerLink
Log in

Quadratic Forms

  • Chapter
  • First Online:
Blocks of Finite Groups and Their Invariants

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2127))

  • 927 Accesses

Abstract

We introduce a quadratic form arising from the Cartan matrix of a block of a finite group. By invoking Brauer’s notion of basic sets, we exploit some properties of the quadratic form with will lead to restriction on the number of characters of the block. We also discuss a question about the indecomposability of Cartan matrices.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

References

  1. Barnes, E.S.: Minkowski’s fundamental inequality for reduced positive quadratic forms. J. Aust. Math. Soc. Ser. A 26(1), 46–52 (1978)

    Article  MATH  Google Scholar 

  2. Brandt, H., Intrau, O.: Tabellen reduzierter positiver ternärer quadratischer Formen. Abh. Sächs. Akad. Wiss. Math.-Nat. Kl. 45(4), 1–261 (1958)

    MathSciNet  Google Scholar 

  3. Brauer, R.: Representations of finite groups. In: Lectures on Modern Mathematics, vol. I, pp. 133–175. Wiley, New York (1963)

    Google Scholar 

  4. Brauer, R.: Some applications of the theory of blocks of characters of finite groups. I. J. Algebra 1, 152–167 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fong, P.: On the characters of p-solvable groups. Trans. Am. Math. Soc. 98, 263–284 (1961)

    MATH  MathSciNet  Google Scholar 

  6. Gantmacher, F.R.: Matrizentheorie. Hochschulbücher für Mathematik, vol. 86. VEB Deutscher Verlag der Wissenschaften, Berlin (1986)

    Google Scholar 

  7. Külshammer, B., Wada, T.: Some inequalities between invariants of blocks. Arch. Math. (Basel) 79(2), 81–86 (2002)

    Google Scholar 

  8. Landrock, P.: A counterexample to a conjecture on the Cartan invariants of a group algebra. Bull. Lond. Math. Soc. 5, 223–224 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  9. Maplesoft, a division of Waterloo Maple Inc.: Maple 16 (2012). http://www.maplesoft.com/products/Maple/

  10. Nebe, G., Sloane, N.: A catalogue of lattices (2014). http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/

  11. Nipp, G.L.: Quaternary Quadratic Forms. Springer, New York (1991). Computer generated tables, With a 3. 5″ IBM PC floppy disk

    Google Scholar 

  12. Olsson, J.B.: Inequalities for block-theoretic invariants. In: Representations of Algebras (Puebla, 1980). Lecture Notes in Mathematics, vol. 903, pp. 270–284. Springer, Berlin (1981)

    Google Scholar 

  13. Sambale, B.: Cartan matrices and Brauer’s k(B)-conjecture. J. Algebra 331, 416–427 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sambale, B.: Cartan matrices and Brauer’s k(B)-Conjecture III. Manuscripta Math. (2014, to appear)

    Google Scholar 

  15. van der Waerden, B.L., Gross, H.: Studien zur Theorie der quadratischen Formen. Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Band 34. Birkhäuser Verlag, Basel (1968)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Benjamin Sambale

Authors
  1. Benjamin Sambale
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Sambale, B. (2014). Quadratic Forms. In: Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, vol 2127. Springer, Cham. https://doi.org/10.1007/978-3-319-12006-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation