Total characters and Chebyshev polynomials

Abstract

The total character τ of a finite group G is defined as the sum of all the irreducible characters of G. K. W. Johnson asks when it is possible to express τ as a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson's question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the first kind in any faithful irreducible character of the dihedral group G

On the Existence of Johnson Polynomials for Nilpotent Groups

“…We call such a polynomial f (x) ∈ Q[x], if it exists, a generalized Johnson polynomial of G. (When we insist that the coefficients are integral, we will say that f (x) is an integral Johnson polynomial.) This problem has been studied for dihedral groups D 2n in [16], where it is proved that D 2n has a generalized Johnson polynomial if and only if 8 n. In [17] and [18], the second author along with R. Sarma and B. Sury, investigated the existence of generalized Johnson polynomials on several classes of p-groups. In all of those cases, they showed that such groups can only have a generalized Johnson polynomial when the group has nilpotence class 2 and a cyclic center.…”

“…Motivated by this, K. W. Johnson raised the following question: does there exist an irreducible character χ of G and a monic polynomial f (x) ∈ Z[x] such that f (χ) = τ G ? (See [16].) If for the group G, there is a monic polynomial f (x) ∈ Z[x] such that f (χ) = τ G , then we say G has a Johnson polynomial.…”