[GAP Forum] Unitary representations

[Laurent Bartholdi]

Dear Dima and Alexander:
Thanks for the quick replies!
Indeed, I was trying to the Cholesky factorization by hand, but GAP can't
extract square roots of cyclotomics, only of rationals. Since the matrices
are 2x2, there's no stability issue; and, to get a unitary representation,
it's crucial to make the invariant scalar product orthonormal, not just
orthogonal (and the field has to go up to CyclotomicField(60) in this case.)

On Tue, Sep 29, 2015 at 1:51 AM, Dima Pasechnik <
dmitrii.pasechnik at cs.ox.ac.uk> wrote:

> On Mon, Sep 28, 2015 at 04:56:47PM -0600, Alexander Hulpke wrote:
> > > Now, classically, I can define s =
> > > Sum(SL(2,5),g->g^rep*TransposedMat(ComplexConjugate(g^rep))), which is
> the
> > > Gram matrix of a positive-definite invariant sesquilinear form; but I
> don't
> > > know how to factor s as t*TransposedMat(ComplexConjugate(t)) so as to
> > > conjugate rep by t.
> >
> > This is a Cholesky decomposition, which (Wikipedia be thanked) seems to
> be a standard operation in the numerical world, and this is exactly what
> GAP uses.
> >
> > However in your concrete example this fails yet — the reason is that the
> Cholesky decomposition requires square roots, and the existing code for
> CholeskyDecomp in GAP cannot deal with square roots for irrationals. (It
> would have to construct a larger field etc. which makes the code far more
> complicated.)
>
> Do you mean the code in pkg/HAPcryst/examples/orbitcoloring.gap ?
>
> In the numerical world people avoid doing the classical Cholesky
> decomposition;
> they do LDL* decomposition (same Wikipedia article),
> which does not need square roots, but leaves you
> with a diagonal sesqulinear form with positive coefficients.
>
> I imagine in some cases it'd just give you a scalar that you can cancel...
>
> Best,
> Dima
>



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Prof. Dr. Laurent Bartholdi   \ laurent.bartholdi<at>gmail<dot>com
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