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

Total characters and Chebyshev polynomials

Division of Natural Sciences, New College of Florida, 5700 NorthTamiami Trail, Sarasota, FL 34243, USA

Received 8 January 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.