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[itex]\epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta}[/itex]

The existence of an invariant of this form (with two indices down) tells us that the covariant (here fundamental of SL(2,C)) and contra variant reps aren't independent. This seems to me to follow directly from the fact that detN=1. Is this true? And can anyone bring forward an example where covariant and contra variant reps are truly independent?