Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A local conjecture on Brauer character degrees of finite groups


Authors: Thorsten Holm and Wolfgang Willems
Journal: Trans. Amer. Math. Soc. 359 (2007), 591-603
MSC (2000): Primary 20C20; Secondary 15A18, 15A36, 16G60, 20C05
DOI: https://doi.org/10.1090/S0002-9947-06-03888-8
Published electronically: July 21, 2006
MathSciNet review: 2255187
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, a new conjecture on the degrees of the irreducible Brauer characters of a finite group was presented by W. Willems. In this paper we propose a `local' version of this conjecture for blocks $ B$ of finite groups, giving a lower bound for $ \sum \varphi(1)^2$ where the sum runs through the set of irreducible Brauer characters of $ B$ in terms of invariants of $ B$. A slight reformulation leads to interesting open questions about traces of Cartan matrices of blocks.

We show that the local conjecture is true for blocks with one simple module, blocks of $ p$-solvable groups and blocks with cyclic defect groups. It also holds for many further examples of blocks of sporadic groups, symmetric groups or groups of Lie type. Finally we prove that the conjecture is true for blocks of tame representation type.


References [Enhancements On Off] (What's this?)

  • 1. R.Brauer, Representations of finite groups. Lectures on Modern Mathematics, John Wiley, New York 1963. MR 0178056 (31:2314)
  • 2. R.Brauer, Notes on representations of finite groups, I. J. London Math. Soc. (2) 13 (1976), 162-166. MR 0399240 (53:3091)
  • 3. P.Brockhaus, On the radical of a group algebra. J. Algebra 95 (1985), 454-472. MR 0801281 (87h:20021)
  • 4. K.Erdmann, Blocks of Tame Representation Type and Related Algebras. Lecture Notes in Mathematics 1428, Springer (1990). MR 1064107 (91c:20016)
  • 5. W.Feit, The representation theory of finite groups. North-Holland, Amsterdam 1982. MR 0661045 (83g:20001)
  • 6. L.Chastkofsky, W.Feit, On the projective characters in characteristic 2 of the groups Suz$ (2^m)$ and Sp$ _4(2^n)$. Inst. Hautes Études Sci. Publ. Math. No. 51 (1980), 9-35. MR 0573820 (83b:20014)
  • 7. B.Huppert, N.Blackburn, Finite groups II, III Springer-Verlag, Berlin, Heidelberg, New York, 1982.
  • 8. B.Huppert, Research in representation theory at Mainz (1984-1990). In: G.O.Michler, C.M.Ringel (eds.), Representation theory of finite groups and finite-dimensional algebras. Birkhäuser, Basel, 1991. MR 1112156 (92c:20011)
  • 9. G.James, A.Kerber, The representation theory of the symmetric group. Encyclopedia of Mathematics and Its Applications 16, Addison-Wesley 1981. MR 0644144 (83k:20003)
  • 10. M.Kiyota, T.Wada, Some remarks on eigenvalues of the Cartan matrix in finite groups. Comm. Algebra 21(11) (1993), 3839-3860. MR 1238129 (94i:20021)
  • 11. P.Landrock, A counterexample to a conjecture on the Cartan invariants of a group algebra. Bull. London Math. Soc. 5 (1973), 223-224. MR 0323887 (48:2240)
  • 12. G.O.Michler, A finite simple group of Lie type has $ p$-blocks with different defects. J. Algebra 104 (1986), 220-230. MR 0866772 (87m:20038)
  • 13. G.Navarro, Characters and Blocks of Finite Groups. Lecture Notes Series 250, Cambridge University Press, Cambridge 1998. MR 1632299 (2000a:20018)
  • 14. W.Schwarz, Die Struktur modularer Gruppenringe endlicher Gruppen der $ p$-Länge $ 1$. J. Algebra 60 (1979), 51-75. MR 0549098 (81a:20012)
  • 15. T.Wada, A lower bound of the Perron-Frobenius eigenvalue of the Cartan matrix for a finite group. Arch. Math. 73 (1999), 407-413. MR 1725175 (2000i:20016)
  • 16. D.A.R.Wallace, On the radical of a group algebra. Proc. Amer. Math. Soc. 12 (1961), 133-137. MR 0121408 (22:12146)
  • 17. W.Willems, On degrees of irreducible Brauer characters. Trans.Am.Math.Soc. 357 (2005), 2379-2387. MR 2140443 (2006a:20019)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C20, 15A18, 15A36, 16G60, 20C05

Retrieve articles in all journals with MSC (2000): 20C20, 15A18, 15A36, 16G60, 20C05


Additional Information

Thorsten Holm
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: Institut für Algebra und Geometrie, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany
Email: tholm@maths.leeds.ac.uk

Wolfgang Willems
Affiliation: Otto-von-Guericke-Universität, Institut für Algebra und Geometrie, Postfach 4120, 39016 Magdeburg, Germany
Email: wolfgang.willems@mathematik.uni-magdeburg.de

DOI: https://doi.org/10.1090/S0002-9947-06-03888-8
Keywords: Brauer character, block of finite group, Cartan matrix, Perron-Frobenius eigenvalue
Received by editor(s): April 25, 2004
Received by editor(s) in revised form: October 28, 2004
Published electronically: July 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society