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Dear Forum,

The following questions of Stefan Kohl already got answered on another

mailing list (By M. Isaacs and B. Howlett). Here is a short summary

and an additional remark for the GAP-forum recipients.

1. Does the character table of a group with n

conjugacy classes only contain

character values which are algebraic of degree

strictly smaller than n ?

(Clearly, this is not a consequence of the fact that

character values of a group G are sums of

Exponent(G)'th roots of unity)

This follows from the following stronger statement:

Let g in G be of order m. Let d be the degree of the field generated

by all character values on g over the rationals. Then d is the number

of conjugacy classes of G represented by powers g^k of g with

gcd(k,m)=1.

Proof: This field is a subfield of K_m = Q[zeta_m], zeta_m a primitive

m-th root of unity. If sigma_k is the automorphism of K_m over Q

mapping zeta_m to zeta_m^k and chi is a character of G then

chi(g)^sigma_k = chi(g^k). The result now follows from basic Galois

theory.

Note that this degree d can be found in GAP without knowing the values

of irreducible characters:

Length(DecomposedRationalClass(RationalClass(G,g)));

or you can read it of from the power map information in the character

table.

2. Let d be the 'determinant' of the character table

of a group G of order n

(in GAP : d := DeterminantMat(List(Irr(G),ValuesOfClassFunction))).

- Is d always different from zero ?

Yes, the irreducible characters are a *basis* of complex class functions.

- Is d^2 always an integer which is divisible by n ?

(Obviously, d is determined up to the sign by the group G,

hence d^2 is uniquely determined by G)

Even d is an integer: It is integral since all matrix entries are

integral and it is rational since applying any field automorphism of

an algebraic closure of the rationals to the table induces a

permutation of the rows. (This also give a direct proof of question

1). If C is the matrix of character values and A is the complex

conjugate transpose of C then the "second orthogonality relations" say

that AC is diagonal with entries the centralizer orders of the

correspondig classes. (Of course, det(AC)=d^2 and the trivial element

has centralizer order |G|.)

With best regards,

Frank

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64, %%% %%% 52064 Aachen, Germany %%% %%% E-mail: Frank.Luebeck@Math.RWTH-Aachen.De %%% %%% Tel: +49-241-80-4549 %%%

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