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askalot

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Suppose we have the diagonal matrix ##P = diag(1,\ldots,1, -1,\ldots, 1)##, with ##N_+## elements of ##1## and ##N_-## elements of ##-1## such as ##N_+ + N_- = N## and ##N_+, N_- \geq 1##.

This matrix is a non trivial parity matrix since it is not proportional to ##I##.

If ##T^\alpha## are the generators of ##SU(N)## and ##[T^\alpha, P] = 0##, I would like to prove that ##SU(N)## breaks as:

##SU(N) \rightarrow SU(N_+) \otimes SU(N_-) \otimes U(1).##

I am particularly interested in the case of ##SU(3) \rightarrow SU(2) \otimes U(1).##

Thanks in advance!