数理解析研究所講究録
第 896 巻 1995 年 8-12

8Sharp characters and their generalizations

Masao KIYOTA

Department of General Education

Tokyo Medical and Dental University

2-8-30 Kohnodai, Ichikawa, Chiba, 272, Japan

1. Blichfeldt’s Theorem

Letof$\chi$.

$G$be a finite group andFor$l\epsilon L$,

a virtual character of$\chi$

we define the number$B(l)= \frac{\alpha(l)}{|G|}$

$G$

. Let

$L$

be the set of values

as follows:$B\langle l)$$\Pi_{l’\in L-\{l\}}(l-l’)$ ,where $a(l)$ denotes the number of elements

in$x$

$G$

with

$\chi(x)=l$.

Ninety years ago, Blichfeldt [B] proved that $B(l)$ is an algebraic integer for any$l\in L$.

Our first aim is to extend this result. We will show that the numbers $B(l)(l\in L)$ are infact the values of a virtual character$\chi$of$G$, constructed from$\chi$in a definite manner.

More precisely, we have the following

Theorem 1.
Then$\tilde{\chi}$Let$\tilde{\chi}$be a class function on

is a virtual character of

$G$

$G$

defined by $\tilde{\chi}(x)=B(\chi(x))$ for $x\in G$ .

.

Since the value of a group character is a sum of roots of unity, it is clear thatTheorem 1 implies Blichfeldt’s Theorem mentioned above.

Proof of Theorem 1. (Outline)

For $x\in G$ , we letdegree whose set of roots is $L-\{\chi(x)\}$ . Let$f= \frac{1}{|G|}\sum_{x\in G}f_{x}$Then vve have the following.

$f$$f_{x}$

denote the monic polynomial of least

be the avarage of$f_{x}$over$G$:

9Claim.

$f$

is a monic polynomial with integral coefficients of degree$|L|-1$ .

In fact, the coefficients of $f$ are expressed by integral linear combinations of 1$i=0,1,\ldots$

and symmetric functions of the elements in$L$.

$i$$\chi$$1_{G}$)For example, if $L=\{n,l,k\}$ thenwe have $f\{X)=X^{2}-(\{n+l+k)-t\chi,1_{G}))X+((nl+lk+kn)-(n+l+k)(\chi.1_{G})+t\chi^{2},1_{G}))$ .

Now Theorem 1 follows easily from Claim sinceRemark.

$\tilde{\chi}=f(\chi)$.

The above $f$ is the polynomial of least degree with $f(l)=B(l)$ for every$l\in L$,that is, the Lagrange interpolation polynomial through the points $\{(l.B(l))|l\in L\}$ .

One of the typical properties ofdefine the class function

Proposition 2.

$-li$$\sim$$(\chi, 1/\chi)=0$

Using Proposition 2

Proposition 3.

$1/\tilde{\chi}$$(\ell i=0)$

For any

integers. In particular, if

is that it does not take the value$0$. So we can. By direct calculation, we obtainfor$i=0,1,\ldots,$$|L|-2$ .

, we have the following divisibility conditions.

$l\in L,$$\chi$$\tilde{\chi}$$B(l)$divides$a(l) \prod_{l’\in L-\{l\}}B\{l’)$is a character of degree$n$in the ring of algebraic, then $B(n)$ divides $\prod_{l\in L-\{n\}}B(l)$ .

2. Sharp characters of finite groups

Under thesame notation as in Section

1, we will define sharp triples for groupcharacters.

Definitions.

The triple $(G, \chi,l)$ is called a sharp triple if $B(l)$ is a unit in the ring ofalgebraic integers.

The pair $(G, \chi)$ is called a sharp pair if $(G, \chi, \chi(1))$ is a sharp triple.

10

The concept of sharp pairs was first introduced by Cameron and Kiyota [CK], andtheir definition of sharp pairs is slightly different from ours. But at least in casea faithful character of$G$$\chi$isare the same. So the concept of sharp, these two definitions

triples is a natural generalization of that of sharp pairs

We will give some examples of sharp triples.

Example 1.

Let$G$sharp for every

Example 2.

be cyclic and$l\in Im\chi$

Let

$G$

be a faithful linear character of G. Then $(G, \chi,l)$ is$\chi$.

be a sharply$t$-transitive permutation group and$\pi$

be the associatedpermutation character. Then $(G, \pi,t-2)$ is a sharp triple, and $(G, \pi)$ is a sharp pair.

The following Lemmas are easy to prove. (Use Proposition 3 for Lemma 5.)

Lemma 4.

If

Lemma 5.

Let$(G, \chi)$$\langle G,$$\chi$$\chi,l)$is sharp, thendivides$\alpha(l)$be a character of degree$n$$|G|$.

. If $(G, \chi,l)$ is sharp for all $l\in L-\{n\}$ , thenis a sharp pair.

Question 6.

If $(G, \chi,l)$ is sharp with$\{x\in G|\chi(x)=l\}$Problem 7.

a faithful character, then is it true that the set$\chi$

is a single conjugacy class ofDetermine all finite groups

irreducible character$\chi$$G$and for every$G$?

such that $(G, \chi,l)$ is sharp for every non-trivial$l\in Im\chi$. Note that abelian groups and dihedralgroups of twice odd prime order are such examples.

3. Classification of sharp triples for given

From

now on we assume$\chi$

isafaithfulcharacterof$L$$G$ofdegree

$n$.

Set

11$L=Im\chi$and $L^{*}=L-\{n\}$ . Cameron and Kiyota [CK] posed the problem of determining all thesharp pairs

for a given set$(G. \chi)$$L^{*}$.

There are many papers on this subject; see theIn particular Alvis and Nozawa [AN] have given areferences of [AKN].

classification of sharp pairs when$L^{*}$

contains an irrational number.

Now we will consider the analogous problem for sharp triples
known to me are very few.

Result 1.

Letsharp, then$G$

Lemma 5. If. The results$\alpha_{t}1$$1’\ldots,$with all$\alpha_{i}$are algebraically conjugate. If$(G, \chi, \alpha_{1})$isis cyclic of prime order.

$t\geqq 2$

by Lemma 4,$(G, \chi,l)$

The first one is the simplest case and easy to prove.

$L^{s}=t\alpha$

Since all$\underline{Proof.}$complete$\alpha_{i}$are conjugate,$(G, \chi, \alpha_{i})$are all sharp, and so $(G, \chi)$ is sharp by, then the result follows from Theorem 4.1 in [CK]. Now assume$\alpha 1\alpha_{1}$divides)$|G|=1+a\langle\alpha_{1})$. Thus$\alpha(\alpha_{1})=1$, andso$G$$t=1$. Thenis cyclic of order

two. This completes the proof.

We will state the other known results without proofs.

Result 2.
$(\chi,1_{G})=0$

Let$L^{*}=\{0,$$\alpha_{1},\ldots,$ $\alpha_{t}I$with alland $(G, \chi,O)$ is sharp, thenorder, or $E$$\lambda Z_{p}$$G$$\alpha_{i}$are algebraically conjugate and$t\geqq 2$. If

is cyclic of order 4, dihedral of twice odd prime, where $p=2^{v}-1$ is a Mersenne prime.

$2^{v}$

Result 3. ( Matsuhisa and Yamaki [MY] )unity. If$(G, \chi,O)$

is sharp, then

$G$

Let

$L^{\iota}=\{0$.

$\epsilon$$1’\ldots,t\epsilon I$

with all$\epsilon_{i}$are roots of

is a sharply 3-transitive group or a 2-transitiveFrobenius group.

Result 4.

Let $L^{*}=\{l,k\}$ with integers $l,k$ . If $t\chi,1_{G}$ ) $=0$ and $(G, \chi,l)$ is sharp, then one of

the following holds:
(i)

$k=0$

and $(G, \chi)$ is sharp of type $\{0,l\}$ .

12is the symmetric group of degree 3.

ii) $k=-1,$ $l=0$

and(iii) $k=-1,$ $l=1$

andProblem 8.

Determine all sharp triples $(G, \chi,l)$ when$\langle$

$G$

$G$

is quaternion or dihedral of order 8.

$L^{*}$

contains an irrational number.

References[AKN] D.Alvis, M.Kiyota and S.Nozawa, On sharp characters of finitegroups, Algebraic

Combinatorics Fukuoka 1993.
[AN]

D.Alvis and S.Nozawa, Sharp characters with irrational values, (submitted).

[B]

H.F.Blichfeldt, A theorem concerning the invariants of linear homogeneous groups,

with some applications to substitution groups, Trans. Amer. Math. Soc. 5 (1904),461-466.
[CK]

P.J.Cameron and M.Kiyota, Sharp characters of finite groups, J. Algebra 115 (1988),

125-143.
[MY]

T.Matsuhisa and H.Yamaki, A class of finite groups admitting certain sharp characters

II, Proc. Amer. Math. Soc. 110, (1990), 1-5.