The Total Character of a Finite Group
Stephen P. Humphries, Chelsea Kennedy, Emma L. Rode · Algebra Colloquium · 2015
The total character τ(G) of a finite group G is the sum of the irreducible characters of G. We present conditions under which τ(G) can be written as a polynomial with integer coefficients in an irreducible character of G. Such a group we call a total character group. We show that the dicyclic group of order 4n is a total character group if and only if n ≡ 2, 3 mod 4. The polynomial used is a sum of Chebyshev polynomials of the second kind. We also show that Sn (n ≥ 4) is not a total character group.