Why does trace play such an effective role?

Question

We know that trace plays a very important role in representation theory in the form of characters or in lie algebra in the form of killing form among many other things. What's so special about trace that makes it such a strong player as opposed to say the sum of a product of eigenvalues taken two at a time, or the sum of entries of the other diagonal (from right up to left down )?

Answer 1

Here are a couple examples of trace occurring without asking for it. Perhaps that will make it seem more special.

The trace map is the derivative of the determinant map $\mbox{det}:\mbox{GL}_n(\mathbb{R}) \to \mathbb{R}$ at the identity, which you would see in studying Lie algebras. For one, it tells you that $\mathfrak{sl}_n(\mathbb{R})$ is the traceless matrices.

It is also the very natural pairing map $V\otimes_\mathbb{R} V^* \to \mathbb{R}$ under the natural identification $\mbox{End}_\mathbb{R}(V) \cong V\otimes_\mathbb{R} V^*$. See here.

(If anyone knows more, please comment and let me know!)