On the Decomposition Numbers of Steinberg's Triality Groups ³D₄(2ⁿ) 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.

ACKNOWLEDGMENTS

Part of this work was done during visits of the first author at the Department of Mathematics of the Chiba University, Japan, and of the National Cheng Kung University, Taiwan. He thanks all persons of these departments for their hospitality and the Japan Society for the Promotion of Science (JSPS) and the National Center for Theoretical Sciences (South) for supporting his visits. The second author acknowledges the support of his research from the National Science Council, Taiwan, Grant no. 100-2115-M-006-007. We thank F. Lübeck for providing a GAP program to compare rational numbers which are given as polynomials in q.

Notes

Communicated by P. Tiep.