Decomposition matrices for low-rank unitary groups

Abstract

We study the decomposition matrices of the unipotent ℓ-blocks of finite special unitary groups SU n ( q ) for unitary primes ℓ larger than n. Up to few unknown entries, we give a complete solution for n = 2 , ... , 10 . We also prove a general result for two-column partitions when ℓ divides q + 1 . This is achieved using projective modules coming from the ℓ-adic cohomology of Deligne–Lusztig varieties.