> < ^ Date: Fri, 20 Jul 2001 16:26:05 -0600 (MDT)
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
> < ^ Subject: Re: Character Value

Dear Gap-Forum,

In the following group and character table , how can I find the value of
X.3 on the element
g:=[ [ Z(2^3)^6, Z(2^3)^6 ], [ Z(2^3)^6, Z(2^3)^5 ] ] in G.


You will (if you like it or not) have to identify the class of the element g.
You can do this in general, by a specific test with \in (but see below):

gap> cl:=ConjugacyClasses(G);;
gap> First([1..Length(cl)],i->g in cl[i]);
4


So you know the class is class number 4, with the name:

gap> ClassNames(CharacterTable(G))[4];
"9a"

So the character value is:

gap> Irr(G)[3][4];
E(9)^2+E(9)^4+E(9)^5+E(9)^7


In this particular case, one can use element orders and the *natural*
character to identify the class of an element (but this does not hold for
arbitrary groups, so we are a bit lucky):

gap> clasprop:=List(ConjugacyClasses(G),
i->[Order(Representative(i)),TraceMat(Representative(i))]);
[ [ 1, 0*Z(2) ], [ 2, 0*Z(2) ], [ 3, Z(2)^0 ], [ 9, Z(2^3) ], [ 9, Z(2^3)^2 ],
[ 9, Z(2^3)^4 ], [ 7, Z(2^3)^5 ], [ 7, Z(2^3)^3 ], [ 7, Z(2^3)^6 ] ]


So we can get the class index from this list:

gap> Position(clasprop,[Order(g),TraceMat(g)]);
4

(GAP will use such invariants automatically, to quickly conclude
that an element does not lie in a certain class.)

If the chosen properties were not unique, and we could not think of another
invariant, we would have to use the (potentially expensive) test via \in.

For more details see the example (from my Diploma thesis) in:

@inproceedings{neubueser93,
author =       "Joachim Neub{\"u}ser",
title =        "An Invitation to Computational Group Theory",
booktitle =    "Groups '93 Galway/St~Andrews",
editor =       "C.~M. Campbell and T.~C. Hurley and E.~F. Robertson
and S.~J. Tobin and J.~J. Ward",
publisher =    CUP,
year =         1995,
pages =        "457--475"
}