The Principal 3-Blocks of Four- and Five-Dimensional Projective Special Linear Groups in Non-defining Characteristic
✍ Scribed by Shigeo Koshitani; Hyoue Miyachi
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 163 KB
- Volume
- 226
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
In representation theory of finite groups, there is a well-known and important conjecture due to M. Broue. He has conjectured that, for any prime p, if a finite ǵroup G has an abelian Sylow p-subgroup P, then the principal p-blocks of G and Ž . the normalizer N P of P in G are derived equivalent. Let q be a power of a G Ž . prime such that q ' 2 or 5 mod 9 . In this paper we show that Broue's conjecture Ž .
Ž . is true for p s 3 and for G s PSL q and G s PSL q . In these cases, G has 4 5 elementary abelian Sylow 3-subgroups of order 9. What we prove here is the Ž . following. In the case G s PSL q all the principal 3-blocks of G are Morita 4 Ž . Ž . even Puig equivalent independently of infinitely many q. In the case G s PSL q 5 Ž . all the principal 3-blocks of G are Morita even Puig equivalent to the principal Ž . 3-block of N P independently of infinitely many q.