The Principal 3-Blocks of Four- and Five-Dimensional Projective Special Linear Groups in Non-defining Characteristic

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He has conjectured that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the principal p-blocks of G and the normalizer NG(P) of P in G are derived equivalent. Let q be a power of a prime such that q ≡ 2 or 5 (mod 9). In this paper we show that Brouéʹs conjecture is true for p = 3 and for G = PSL4(q) and G = PSL5(q). In these cases, G has elementary abelian Sylow 3-subgroups of order 9. What we prove here is the following. In the case G = PSL4(q) all the principal 3-blocks of G are Morita (even Puig) equivalent independently of infinitely many q. In the case G = PSL5(q) all the principal 3-blocks of G are Morita (even Puig) equivalent to the principal 3-block of NG(P) independently of infinitely many q.