Total Characters of Dihedral Groups and Sharpness

Abstract

We define the total character $\tau$ of a finite group $G$ as the sum of all its irreducible characters. A question of K. W. Johnson asks whether the total character of a finite group can be expressed as a polynomial with integer coefficients in some irreducible character $\chi$ of $G$. We show that in the case of dihedral groups of twice odd order the question has an affirmative answer and we give the explicit polynomial.