[GAP Forum] Unitary representations

[Alexander Hulpke]

Dear Laurent, Dear Forum,


In principle (in recent versions of GAP, this got added later), adding the option `unitary’, i.e.

IrreducibleRepresentations(SL(2,5):unitary);

will produce unitary representations.

> 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.)

Anyhow, this might tell you how to do it by hand.

Best,

  Alexander

> 
> Any ideas?
> 
> Thanks in advance, Laurent
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