importance of invariant tensors while studying representations of SL(2,C), for raising and lowering indices of spinors invariant tensor $\epsilon$ was constructed analogous to $\eta$ in SO(1,3).What is the importance of invariant tensors? Are they significant only because they are helpful in raising and lowering indices?
 A: That's a result of classical 19th century invariant theory called the first fundamental theorem for $SL(n,\mathbb{C})$. See for example this MO answer for a proof and more background. All polynomial invariants of a bunch of tensors are obtained by contracting different indices (upper with lower) directly or similar indices through the mediation of an $\epsilon$. The latter is the tensor corresponding to the determinant which defines the group $SL$ (the same way a quadratic form defines an orthogonal group as its group of invariance). This is the reason it plays such an important role in the representation/invariant theory of the group $SL$. Note that $n=2$ is special: representations are self-dual and that translates into the statement that you can always raise or lower indices with an $\epsilon$ which is a matrix. For arbitrary $n$, the epsilon is a tensor with $n$ indices. You can still use it to raise or lower indices but then each say lower index that you raise will produce $n-1$ upper indices and this will completely change the format of the tensors involved.
