We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study how the unitriangular basic sets of the alternating and the symmetric groups are related and obtain several results of existence. We show that these sets do not always exist in the case of the alternating groups by studying two explicit cases in characteristic 3. We then consider the case of a symmetric algebra and we show how one can obtain unitriangular basic sets in this more general context.
The first and third authors acknowledge the support of the ANR grant GeRepMod ANR-16-CE40-0010-01. The third author is also supported by ANR project JCJC ANR-18-CE40-0001. The authors sincerely thank Gunter Malle for his precise reading of the paper and his helpful comments. Last, the authors warmly thank the referee for a lot of helpful comments and corrections on the paper.
- Basic set
- Decomposition matrix of the alternative groups
- decomposition matrix of the Hecke algebras