TOTAL CHARACTERS AND CHEBYSHEV POLYNOMIALS

PII. S0161171203201046 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

TOTAL CHARACTERS AND CHEBYSHEV POLYNOMIALS

EIRINI POIMENIDOU and HOMER WOLFE Received 8 January 2002

The total characterτof a finite groupGis defined as the sum of all the irreducible characters ofG. K. W. Johnson asks when it is possible to expressτas a polyno- mial 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 groupG.

2000 Mathematics Subject Classification: 20C15.

1. Introduction. AGel’fand modelMfor a groupGwas defined in [8] as any complex representation ofGwhich is isomorphic to the direct sum of all irre- ducible representations ofG. We refer to the character of such a representation as thetotal characterτofG. A Gel’fand model afforded by a (generalized) per- mutation representation is referred to as a (weakly) geometric Gel’fand model.

When a Gel’fand modelMis the representation afforded by a nonnegative inte- ger linear combination of powers of a genuineG-setX, then the total character τ ofGcan be expressed as a polynomial in the characterχafforded byX. A question related to this idea was posed by Johnson [5].

Question1.1. For a finite groupG, do there necessarily exist an irreducible characterχand a monic polynomialf (x)∈Z[x]such thatf (χ)=τ, whereτ is the total character ofG?

Johnson’s question arose in the context of character sharpness which we will briefly explain. LetGbe a finite group,χa generalized character ofGof degree n,L= {χ(g)|g=1}, andfL(x)=

l∈L(x−l). It was discovered by Blichfeldt [2] and rediscovered in a modern context by Kiyota [6] thatf_L(x)∈Z[x]and

|G|dividesf_L(n). In the special case thatf_L(n)= |G|, the characterχis said to besharp. Another way to characterize a sharp character is to notice that the class functionf_L(χ)=

l∈L(χ−l1_G)=ρ, whereρis the regular character and 1_Gis the trivial character ofG. In other words, every irreducible character of Gappears as a constituent offL(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.

Theorem1.2. LetGD2nand letτbe the total character ofG.

(1) If n is odd or ifn=2m, where m≡2,3(mod 4), then there exists a unique monic polynomialP(x)∈Z[x]such thatP(χ)=τfor any faithful χ∈Irr(G). Moreover,

P(x)=





























 2

m k=0

T_kx 2

−2T_m−1

x 2

, ifn=2m−1, n≡1,3(mod 8),

2

m−1 k=0

Tk

x 2

, ifn=2m−1, n≡5,7(mod 8),

2 m k=0

T_k x

2

, ifn=2m, m≡2,3(mod 4),

(1.1)

whereT_k(x)is thekth Chebyshev polynomial of the first kind.

(2) Ifn=2mandm≡1(mod 4), then there exists a unique monic polyno- mialP(x)∈Z[x]such thatP(χ)=2τfor any faithfulχ∈Irr(G), where

P(x)=2



T_m+1 x

2

+2 m k=0

Tk

x 2

−T_m−1 x

2

. (1.2)

Moreover, there does not existP(x)∈Z[x]such thatP(χ)=τ for any χ∈Irr(G).

(3) IfGD2n, wheren≡0(mod 8), then there does not exist a polynomial P(x)∈C[x]such thatP(χ)=τfor anyχ∈Irr(G).

2. Preliminaries. The total character for all dihedral groups was computed in [7, Proposition 2.1]. A dihedral group of order 2nwithn≥3 will be pre- sented as usual as

D_2n=G=a,b:aⁿ=b²=1, b⁻¹ab=a⁻¹. (2.1)

Using the notation in [4], we use g_i and h_i to denote a representative and the size of theith conjugacy class, respectively. The character table and total characterτofD2nare given below.

Case1(nodd). The conjugacy classes ofD2n(nodd) are

{1},

a^r,a^−r 1≤r≤n−1 2

,

a^sb|0≤s≤n−1

. (2.2)

The character table ofD_2n(nodd) and the total characterτ, where=e^2πi/n, is presented inTable 2.1.

Table2.1

g_i 1 a^r

1≤r≤(n−1)/2 b

h_i 1 2 n

χ1 1 1 1

χ₂ 1 1 −1

ψ_j

1≤j≤(n−1)/2

2 ^jr+^−jr 0

τ n+1 1 0

Case2(neven). Ifnis even, sayn=2m, then the conjugacy classes ofD_2n are

{1}, a^m

,

a^r,a^−r

(1≤r≤m−1),

a^sb|seven, a^sb|sodd. (2.3)

The character table ofD_2n (neven,n=2m, and the total characterτ, where =e^2πi/n) is given inTable 2.2.

Table2.2

g_i 1 a^m a^r(1≤r≤m−1) b ab

h_i 1 1 2 n/2 n/2

χ1 1 1 1 1 1

χ₂ 1 1 1 −1 −1

χ₃ 1 (−1)^m (−1)^r 1 −1

χ3 1 (−1)^m (−1)^r −1 1

ψ_j(1≤j≤m−1) 2 2(−1)^{j jr}+^−jr 0 0

τ 2(m+1) 0, modd, 0, rodd,

0 0

2, meven, 2, reven,

The nth Chebyshev polynomial of the first kind is defined as T_n(x) = cos(ncos⁻¹(x))for|x| ≤1. Chebychev polynomials can be expressed recur- sively asT_n+2(x)=2xT_n+1(x)−Tn(x), T0(x)=1, andT1(x)=x. Before we proceed with the proof ofTheorem 1.2we need the following lemmas.

Lemma2.1. IfP(ψ1)=τ for some P(x)∈Z[x], then P(ψj)=τ for any faithful characterψ_jofD_2n.

Proof. We first observe thatψ_j is faithful if and only if (j,n)=1, for ^jr+^−jr =2 cos(2πr j/n)=2 if and only ifn|r j if and only if (n,j)=1.

Hence for(j,n)=1, the character ψj is faithful and a Galois conjugate of ψ1with the same set of character values. Since fornoddP(ψ1[a^r])=1 for all r, it follows that P(ψ_j[a^r])=P(ψ1[a^r])=1. When nis even, we have thatP(ψ1[a^r])=2 whenr is even andP(ψ1[a^r])=0 whenr is odd. For a faithfulψj, the set of character values is the same as those ofψ1; and sincejis necessarily odd, it follows thatP(ψ_j[a^r])=2 whenris even, andP(ψ_j[a^r])= 0 whenr is odd as required.

Lemma2.2. Let thenth Chebyshev polynomial of the first kind be expressed asTn(x)=n

k=1cn,kx^k+cn,0. Thencn,n=2⁽ⁿ⁻¹⁾and2^(k−1)|cn,kfor1≤k≤n.

Proof. The first half follows easily by an inductive argument onn and by using the recursive relation mentioned above. For divisibility of the coeffi- cients, observe that the result is true forn=1 andn=2. Now assume that the result is true for all Chebyshev polynomials of degree less thatnand consider thenth degree Chebyshev polynomialT_n(x). We have by the recursive relation thatc_n,k=2c_n−1,k−1−c_n−2,k. By inductive hypothesis, 2^(k−1)|2c_n−1,k−1−c_n−2,k for any 1≤k≤n. Hence the result is true for all the coefficients ofTn(x)and hence for alln.

Lemma2.3[3, 134.2]. Form∈Z⁺, m

k=0

coskx=1 2

1+sin

(m+1/2)x sin

(1/2)x

. (2.4)

Lemma2.4. Letpm(x)=2m

k=0Tk(x/2). The following equations hold:

p_m(2)=2(m+1); (2.5)

p_m

2 cos2πr 2m

=





0, ifrodd,

2, ifreven; (2.6)

p_m 2 cos

2πr (2m−1)

=1+(−1)^r2 cos πr

(2m−1)

; (2.7)

p_m(0)=1+sin mπ

2

+cos mπ

2

. (2.8)

Proof. Equation (2.5) follows immediately sinceT_k(1)=1, for all k≥0.

For the other equations, we useLemma 2.3,

p_m 2 cos

2πr 2m

=2 m k=0

cos πkr

m

=1+sin(m+1/2)(πr /m) sin(πr /2m)

=1+cos(πr )=





0, ifrodd, 2, ifreven,

p_m 2 cos

2πr 2m−1

=1+sin(2m−1)+2/22πr /(2m−1) sin

πr /(2m−1)

=1+cos(πr )sin 2

πr /(2m−1) sin

πr /(2m−1)

=1+(−1)^r2 cos πr

2m−1

,

p_m(0)=2 m k=0

T_k(0)=2 m k=0

cos kπ

2

=1+sin

(m+1/2)(π/2) sin(π/4)

=1+sin mπ

2

+cosmπ 2

.

(2.9)

3. Proof of main theorem. It is a well-known theorem of polynomial inter- polation that there exists a unique polynomial of degreekor 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 de- gree less than the number of distinct values of someχ∈Irr(G), which matches the values of the total character when considered as a class function onχ, we immediately get that the polynomial is both minimal and unique.

Proof ofTheorem1.2(1). Letn=2m−1,n≡1,3(mod 8). UsingLemma 2.4, we have thatP(x)=pm(x)−2T_m−1(x/2),

Pψ1[1]

=2(m+1)−2=2m=n+1=τ[1], P

ψ1

a^r

=1+(−1)^r2 cos πr

2m−1

−2 cos

πr (m−1) 2m−1

=1=τa^r, P

ψ1[b]

=1+sin mπ

2

+cos mπ

2

−2=0=τ[b].

(3.1)

Letn=2m−1,n≡5,7(mod 8). ThenP(x)=p_m−1(x), P

ψ1[1]

=

2(m−1)+1

=2m=n+1=τ[1],

P ψ1

a^r

=2

m−1 k=0

cos 2πkr

2m−1

=1+ sin(πr ) sin

πr /(2m−1)=1=τa^r, Pψ1[b]

=1+sin mπ

2

+cos mπ

2

=0=τ[b].

(3.2)

Letn=2m,m≡2,3(mod 4). ThenP(x)=pm(x), P

ψ1[1]

=2(m+1)=τ[1],

P ψ1

a^r

=





0, ifrodd, 2, ifreven,

=τa^r, Pψ1[b]

=Pψ1[ab]

=1+sin mπ

2

+cos mπ

2

=0

=τ[b]=τ[ab].

(3.3)

Thus, in each of the above three cases,P(x)is minimal, and byLemma 2.2it is monic with integer entries. ByLemma 2.1, we have thatP(ψ_j)=τfor any faithfulψ_j∈Irr(G).

Proof ofTheorem1.2(2). Letn=2mand m≡1(mod 4). ThenP(x)= 2T_m+1(x/2)+2p_m(x)−2T_m−1(x/2),

Pψ1[1]

=2

1+2(m+1)−1

=2τ[1],

Pψ1a^r

=2

cos(m+1)πr

m +1+(−1)^r−cos(m−1)πr m

=





0, ifrodd, 4, ifreven,

=2τa^r, P

ψ1[b]

=P

ψ1[ab]

=2

0+

1+sin mπ

2

+cos mπ

2

−0

=0=2τ[b]=2τ[ab].

(3.4)

Again, we have thatP(x) is minimal since it matchesm+2 distinct values with a polynomial of degreem+1. ByLemma 2.1,P(ψ_j)=2τfor any faithful character ofG, and byLemma 2.2it is monic with integer entries. Thus, the unique polynomial that maps the values of a faithful χ∈Irr(G) ontoτ is (1/2)P(x)and has noninteger coefficients.

Proof ofTheorem1.2(3). Letn=2mandm≡0(mod 4). Whenr=m/2, we have that for any faithful character,ψj[a^m/2]=ψj(b)=0, butτ[a^m/2]=2 andτ(b)=0; thus we have an inconsistent system.

Acknowledgments. Both authors wish to thank Professor Donald Col- laday of New College and Professor Kenneth Johnson of Pennsylvania State University for their comments and suggestions during writing this paper. We

also wish to thank the New College Foundation for supporting this project through a faculty/student development grant.

References

[1] A. Björck and G. Dahlquist,˙ Numerical Methods, Prentice-Hall Series in Automatic Computation, Prentice-Hall, New Jersey, 1974.

[2] H. F. Blichfeldt, A theorem concerning the invariants of linear homogeneous groups, with some applications to substitution-groups, Trans. Amer. Math.

Soc.5(1904), no. 4, 461–466.

[3] I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, Academic Press, Massachusetts, 1994.

[4] G. James and M. Liebeck,Representations and Characters of Groups, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1993.

[5] K. W. Johnson, private correspondence, 1997.

[6] M. Kiyota,An inequality for finite permutation groups, J. Combin. Theory Ser. A 27(1979), no. 1, 119.

[7] E. Poimenidou and A. Cottrell,Total characters of dihedral groups and sharpness, Missouri J. Math. Sci.12(2000), no. 1, 12–25.

[8] J. Soto-Andrade,Geometrical Gel’fand models, tensor quotients, and Weil represen- tations, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., vol. 47, American Mathematical So- ciety, Rhode Island, 1987, pp. 305–316.

Eirini Poimenidou: Division of Natural Sciences, New College of Florida, 5700 North Tamiami Trail, Sarasota, FL 34243, USA

E-mail address:[email protected]

Homer Wolfe: Division of Natural Sciences, New College of Florida, 5700 North Tami- ami Trail, Sarasota, FL 34243, USA

E-mail address:[email protected]