> < ^ Date: Fri, 23 Jun 2000 12:19:23 +0200 (CEST)
> < ^ From: Frank Luebeck <frank.luebeck@math.rwth-aachen.de >
> < ^ Subject: Re: Some questions about character tables

Dear Forum,

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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