Total characters and Chebyshev polynomials.
Summary: The total character \(\tau\) of a finite group \(G\) is defined as the sum of all the irreducible characters of \(G\). K. W. Johnson asked when it is possible to express \(\tau\) 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 \(G\).
scientific article; zbMATH DE number 2012549