In the study of representation theory of the (complexified) Lie algebras su($N$) and so($N$), one often constructs the tensor product of their defining representations, and then reduces it to smaller irreducible representations by taking traces or symmetrizing/anti-symmetrizing the indices of the tensors that "realize" the representation.
But how is the (anti-)symmetrization step motivated? Why are (anti-)symmetric tensors important and related to irreducible representations of su($N$) and so($N$)? Can one please give a relatively intuitive and heuristic explanation?
Edit: Comments state that the symmetrization is motivated by the fact that the symmetric (permutation) group acts naturally on the tensor product space, and this action permutes with the Lie group action. Then the eigenspaces of a certain permutation naturally decompose the tensor representation as the direct sum of smaller ones. But to my knowledge we are only interested in eigenspaces of the permutations with eigenvalue $+ 1$ (symmetric tensors) and $-1$ (anti-symmetric tensors). Why eigenspaces with complex eigenvalues $e^{2\pi i/n}$ are not considered?