© Hindawi Publishing Corp. TOTAL CHARACTERS AND CHEBYSHEV POLYNOMIALS

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 .


Introduction.
A Gel'fand model M for a group G was defined in [8] as any complex representation of G which is isomorphic to the direct sum of all irreducible representations of G.We refer to the character of such a representation as the total character τ of G.A Gel'fand model afforded by a (generalized) permutation representation is referred to as a (weakly) geometric Gel'fand model.When a Gel'fand model M is the representation afforded by a nonnegative integer linear combination of powers of a genuine G-set X, then the total character τ of G can be expressed as a polynomial in the character χ afforded by X.A question related to this idea was posed by Johnson [5].
Question 1.1.For a finite group G, do there necessarily exist an irreducible character χ and a monic polynomial f (x) ∈ Z[x] such that f (χ) = τ, where τ is the total character of G?
Johnson's question arose in the context of character sharpness which we will briefly explain.Let G be a finite group, χ a generalized character of G of degree n, L = {χ(g) | g = 1}, and f L (x) = l∈L (x − l).It was discovered by Blichfeldt [2] and rediscovered in a modern context by Kiyota [6] In the special case that f L (n) = |G|, the character χ is said to be sharp.Another way to characterize a sharp character is to notice that the class function f L (χ) = l∈L (χ − l1 G ) = ρ, where ρ is the regular character and 1 G is the trivial character of G.In other words, every irreducible character of G appears as a constituent of f L (χ).
A partial answer to Johnson's question was given in [7] for certain dihedral groups.In this paper, we give a complete treatment for all dihedral groups and we show that the right polynomials, when they exist, are integer sums of Chebyshev polynomials of the first kind.
We prove the following main theorem.Theorem 1.2.Let G D 2n and let τ be the total character of G.

Preliminaries.
The total character for all dihedral groups was computed in [7,Proposition 2.1].A dihedral group of order 2n with n ≥ 3 will be presented as usual as (2.1) Using the notation in [4], we use g i and h i to denote a representative and the size of the ith conjugacy class, respectively.The character table and total character τ of D 2n are given below.
Case 1 (n odd).The conjugacy classes of D 2n (n odd) are The character table of D 2n (n odd) and the total character τ, where = e 2πi/n , is presented in Table 2.1. 3) The character table of D 2n (n even, n = 2m, and the total character τ, where The nth Chebyshev polynomial of the first kind is defined as T n (x) = cos(n cos −1 (x)) for |x| ≤ 1. Chebychev polynomials can be expressed recursively as T n+2 (x) = 2xT n+1 (x) − T n (x), T 0 (x) = 1, and T 1 (x) = x.Before we proceed with the proof of Theorem 1.2 we need the following lemmas.Proof.We first observe that ψ j is faithful if and only if (j, n) = 1, for jr + −jr = 2 cos(2πr j/n) = 2 if and only if n|r j if and only if (n, j) = 1.
Hence for (j, n) = 1, the character ψ j is faithful and a Galois conjugate of ψ 1 with the same set of character values.Since for n odd P (ψ 1 [a r ]) = 1 for all r , it follows that P (ψ j [a r ]) = P (ψ 1 [a r ]) = 1.When n is even, we have that P (ψ 1 [a r ]) = 2 when r is even and P (ψ 1 [a r ]) = 0 when r is odd.For a faithful ψ j , the set of character values is the same as those of ψ 1 ; and since j is necessarily odd, it follows that P (ψ j [a r ]) = 2 when r is even, and P (ψ j [a r ]) = 0 when r is odd as required.
Lemma 2.2.Let the nth Chebyshev polynomial of the first kind be expressed as Proof.The first half follows easily by an inductive argument on n and by using the recursive relation mentioned above.For divisibility of the coefficients, observe that the result is true for n = 1 and n = 2. Now assume that the result is true for all Chebyshev polynomials of degree less that n and consider the nth degree Chebyshev polynomial T n (x).We have by the recursive relation that Hence the result is true for all the coefficients of T n (x) and hence for all n. (2.9)

Proof of main theorem.
It is a well-known theorem of polynomial interpolation that there exists a unique polynomial of degree k or less that maps k + 1 distinct points in the domain to predefined points in the range.We refer the reader to [1] for a natural proof.Thus, by presenting a polynomial of degree less than the number of distinct values of some χ ∈ Irr(G), which matches the values of the total character when considered as a class function on χ, we immediately get that the polynomial is both minimal and unique.

Proof of Theorem 1.2(2).
Let n = 2m and m ≡ 1(mod 4).Then Again, we have that P (x) is minimal since it matches m + 2 distinct values with a polynomial of degree m + 1.By Lemma 2.1, P (ψ j ) = 2τ for any faithful character of G, and by Lemma 2.2 it is monic with integer entries.Thus, the unique polynomial that maps the values of a faithful χ ∈ Irr(G) onto τ is (1/2)P (x) and has noninteger coefficients.
Proof of Theorem 1.2(3).Letn = 2m and m ≡ 0(mod 4).When r = m/2, we have that for any faithful character, ψ j [a m/2 ] = ψ j (b) = 0, but τ[a m/2 ] = 2 and τ(b) = 0; thus we have an inconsistent system.also wish to thank the New College Foundation for supporting this project through a faculty/student development grant.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: