Singer cycles in 2-modular representations of the group $$GL_{n+1}(2)$$ G L n + 1 ( 2 )

Abstract

We determine the irreducible 2-modular representations of the group $$G=GL_{n+1}(2)$$ in which a Singer cycle has eigenvalue 1, and show that in these representations every element $$g\in G$$ has eigenvalue 1.