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How does one go about finding a matrix, U, such that U^{-1}D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
What bugs me is how to get block-diagonal matricies if your representation is ##\mathbf{D}_1\oplus \mathbf{D}_1##, or something similar, i.e. if you have two or more copies of the same representation. The projection operators will not touch it.Finally, represent ##\mathbf{M}## in that eigenvector basis for all ##g\in G## (i.e. ##\mathbf{U}## that you wanted consists of these eigenvectors). This will be block-diagonal.
How is it given, if not already in block form, i.e. how do you know, that the two spaces are invariant?What bugs me is how to get block-diagonal matricies if your representation is ##\mathbf{D}_1\oplus \mathbf{D}_1##, or something similar, i.e. if you have two or more copies of the same representation. The projection operators will not touch it.
Does anyone know?
It is motivated by my earlier post (see above) , but it is not too important.How is it given, if not already in block form, i.e. how do you know, that the two spaces are invariant?
That's my question, how do I find such basis for this specific situation, i.e. where I know that representation is a direct sum of two copies of the same irrep, but I do not know in which basis? What is the procedure?If they are, find a basis for both and perform the change of basis on your matrices.
This is not necessarily true. It may be written in a form which is rotated away from the block form and still be a ##\mathbf{D}\oplus\mathbf{D}## representation, i.e., it may be written in a basis that mixes the irreps. This would not mean that it is not a ##\mathbf{D}\oplus\mathbf{D}## representation.I haven't checked your statements, but given they are correct, why isn't ##\mathbf{U}^\dagger \mathbf{M}\mathbf{U}## the matrix representation you are looking for? If it really equals ##\mathbf{D}\oplus \mathbf{D}## then it is in block form.
But he wrote the ##\mathbf{D}## as ##2\times 2## matrices and ##\mathbf{M}## as ##4\times 4##!This is not necessarily true. It may be written in a form which is rotated away from the block form and still be a ##\mathbf{D}\oplus\mathbf{D}## representation, i.e., it may be written in a basis that mixes the irreps. This would not mean that it is not a ##\mathbf{D}\oplus\mathbf{D}## representation.
Precisely. The rules of the 'game' is that you are given ##\mathbf{M}\left(g\right)##'s for each ##g\in G## and you are told what ##G## is, and ##G## is finite. ThenThis is not necessarily true. It may be written in a form which is rotated away from the block form and still be a D⊕D\mathbf{D}\oplus\mathbf{D} representation
How would I find ##\mathbf{U}## if I was not given it?
look for invariants and start with all semisimple parts.