[GAP Forum] Sqrt for the cyclotomic numbers

[Dima Pasechnik]

On Tue, Jan 20, 2015 at 07:31:56AM +0000, Palcoux Sebastien wrote:
> Dear Alexander and Forum,
> If the cyclotomic number is the square of a cyclotomic number, is there an easy way to find it?
> The number I need are the eigenvalues of the matrix of the unitarized inner product of an irreducible representation of a finite group (see the comment of Paul Garett here: http://math.stackexchange.com/q/1107941/84284). This matrix is positive, I guess its eigenvalues are always cyclotomic (true for the examples I've looked, but I don't know in general), and I hope they are square of cyclotomic. Thanks to these square roots I can compute the unitary matrices for the irreducible representation.

You don't need to take square roots. If H is the Hermitian positive definite form
you obtained by the averaging (or in some other way) then H=LDL*, for 
L a lower-triangular matrix with 1s on the main diagonal, and D is a diagonal matrix.
L and D can be computed without taking square roots (and so they will stay cyclotomic). 
Then conjugating by L gives you the unitary form.

HTH,
Dmitrii


> Remark: a function on GAP computing the unitary irreducible representations seems very natural, so if there is not such a function, this should means that there are problems for computing them in general with GAP, isn't it?
> Best regards,Sebastien Palcoux        
> 
>      Le Mardi 20 janvier 2015 3h13, Alexander Hulpke <hulpke at fastmail.fm> a écrit :
>    
> 
>  Dear Forum,
> 
> > On Jan 19, 2015, at 1/19/15 2:18, Palcoux Sebastien <sebastienpalcoux at yahoo.fr> wrote:
> > 
> > Hi,
> > Is it possible to extend the function Sqrt on the cyclotomic numbers?
> 
> How would you represent this root? In general the square root of a cylotomic is not cyclotomic again. (You could form a formal AlgebraicExtension, but then you lose the irrational cyclotomics for operations.)
> 
> Regards,
> 
>   Alexander Hulpke