On the Decomposition Numbers of Steinberg's Triality Groups 3 D 4(2 n ) in Odd Characteristics

Abstract

We determine the ℓ-modular decomposition matrices of Steinberg's triality groups 3 D 4(2 n ) for all primes ℓ ≠ 2 except for some entries in the unipotent characters. As an application, we classify all absolutely irreducible representations of 3 D 4(2 n ) in non-defining characteristic up to a certain degree.