International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 38, Pages 2447-2453
doi:10.1155/S0161171203201046

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.