Problems on Brauer characters

Document Type : Research Paper

Authors

1 School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China

2 Fakultat fur Mathematik, Otto-von-Guericke Universitat, Magdeburg, Germany and Departamento de Matematicas Universidad del Norte, Barranquilla, Colombia

3 School of Mathematical Sciences, Peking University, Beijing, China

Abstract

In this paper we focus on problems of irreducible $p$-Brauer characters and state conjectures based on many examples which we computed. Many of the asked questions hold true for $p$-solvable groups, but their answers in general seem to require a much deeper understanding than we have at the moment. The questions are dealing with degrees, Hilbert divisors, divisibility, height-zero irreducible Brauer characters, number of irreducible Brauer characters in a $p$-block, and Cartan invariants. For instance, one of the main conjectures states that an irreducible Brauer character has Hilbert divisor 1 if and only if the character lies in a $p$-block of defect zero. This is true for $p$-solvable groups since in this case there is a relation between Hilbert divisors and vertices. However, even for a $p$-block of a non-$p$-solvable group containing only two irreducible Brauer characters we do not have an idea how to attack the problem. We hope that the conjectures and questions which we state in this paper will inspire further research.

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References

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International Journal of Group Theory
Volume 15, Issue 4 - Serial Number 4
December 2026
Pages 179-192

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History

  • Receive Date: 28 October 2025
  • Revise Date: 02 December 2025
  • Accept Date: 13 December 2025
  • Published Online: 17 December 2026

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