A tensor square theorem for characters of ${\mathrm{GL}}_n(q)$

Abstract

We construct an irreducible character of ${\mathrm{GL}}_n(q)$ whose square contains every irreducible character that is trivial on the centre of ${\mathrm{GL}}_n(q)$. This character is related to, but in general not equal to, the Steinberg character. A key ingredient, which we also prove, is that every nonlinear irreducible character of ${\mathrm{GL}}_n(q)$ that is trivial on the centre contains the trivial character when restricted to the diagonal subgroup $T$.