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I have a question on G-parity. I know it's defined as ## G = exp(-i\pi I_{y})C ##, with ##I_y## being the second component of the isospin and ##C## is the C-parity. In other words, the G-parity should be the C-parity followed by a 180° rotation around the second axis of the isospin.

Now, if I apply C on the isospin doublet ## \begin{pmatrix} u\\ d \end{pmatrix} ## I get ## \begin{pmatrix} \bar{u}\\ \bar{d} \end{pmatrix} ##, which is quite clear.

If I apply the G-parity on the same doublet I get: ## -\begin{pmatrix} \bar{d}\\ \bar{u} \end{pmatrix} ##. I understand ##\bar{d}## and ##\bar{u}## are now inverted because of the rotation around ##I_y##, but where does the minus sign come from?