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kalmarek / SymbolicWedderburn.jl

Amazing package to compute decompositions into irreducibles of explicit group representations and the Wedderburn decomposition for endomorphisms thereof!
MIT License
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Non-orthogonal semisimple summand #35

Closed blegat closed 3 years ago

blegat commented 3 years ago

With the group in https://gist.github.com/kalmarek/37fde1975162b9709f72d6a312fcbf91, the basis is

1 0 -1 0
0 1 -1 0

which gives the induced matrices

-1 -1
 1  0

and

 1  0
-1 -1

which are not orthogonal. The SDP solution is

 2 -1
-1  2

which has eigenvalues (1, 3) and eigenvectors (1, 1) and (1, -1) so we guess a different basis

1  1 -2 0
1 -1  0 0

which has orthogonal induced matrices. Would it be possible for SymbolicWedderburn to return this basis instead ?

kalmarek commented 3 years ago

orthogonal bases are not really possible with exact/symbolic numbers; for now if you specify Float64 (or Complex64) as the first argument to symmetry_adapted_basis svd will be used to find an orthogonal basis, otherwise we keep it as is (see the docstring for symmetry_adapted_basis).