taking derivatives componentwise and also it’s an invariant tensor of SU(N)
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No. Special linear is matrices with unit determinant. How do we write the determinant of matrix M_{ij} in index notation? det(M) = epsilon^{i_1 i_2 ... i_n} M_{1 i_1} M_{2 i_2} ... M_{n i_n}. You need the epsilon to do this -- it is the invariant symbol.
good point i thought for some reason the proof involved unitarity