Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

Original · EN

Let G be a finite simple group of Lie type, and let πG be the permutation representation of G associated with the action of G on itself by conjugation. We prove that every irreducible representation of G is a constituent of πG, unless G=PSUₙ(q) and n is coprime to 2(q+1), where precisely one irreducible representation fails. Let St be the Steinberg representation of G. We prove that a complex irreducible representation of G is a constituent of the tensor square St⊗ St, with the same exceptions as in the previous statement.