[GAP Forum] Unitary representations

[Dima Pasechnik]

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