My experience with index notation is strictly in the context of special relativity where the real vectorspace $V$ is equipped with a symmetric, non-degenerate bilinear form $g:V\times V\to \mathbb{R}$. However, in reading a book on Group theory in physics, I have come across the following strange conventions (see, for example, pg 293 of Group Theory in Physics by Wu-ki Tung)

$x_i^\dagger=\bar{x^i}$

$D^{\dagger}\ ^{i}\ _j =\bar{D^j}_i $

which make me question how the rules of the formalism need to be modified.

In particular, the whole concept of raising and lowering of indices relies on the existence of linear (musical) isomorphisms between $V$ and $V^*$. For a complex vectorspace equipped with a nondegenerate Hermitian form, these are no longer linear and the bilinear form no longer symmetric. How do the usual rules like $$g(V,W)=V^\alpha W_\alpha $$ $$T^i\ _j=g^{ik} T_{kj}$$ need to be changed?

I would be greatful if someone can point me to a source that explicitly explains this.

My experience with index notation is strictly in the context of special relativity where the real vector space $V$ is equipped with a symmetric, non-degenerate bilinear form $g:V\times V\to \mathbb{R}$. However, in reading a book on Group theory in physics, I have come across the following strange conventions (see, for example, pg 293 of Group Theory in Physics by Wu-ki Tung)

$x_i^\dagger=\bar{x^i}$

$D^{\dagger}\ ^{i}\ _j =\bar{D^j}_i $

which make me question how the rules of the formalism need to be modified.

In particular, the whole concept of raising and lowering of indices relies on the existence of linear (musical) isomorphisms between $V$ and $V^*$. For a complex vector space equipped with a non-degenerate Hermitian form, these are no longer linear and the bilinear form no longer symmetric. How do the usual rules like $$g(V,W)=V^\alpha W_\alpha $$ $$T^i\ _j=g^{ik} T_{kj}$$ need to be changed?

I would be grateful if someone can explains this.