I have found two different papers (both are well cited) Classification of Self-Orthogonal Codes over $\mathbb{F}_3$ and $\mathbb{F}_4$ (pg.3) and Convolutional and Tail-Biting Quantum Error-Correcting Codes (end of pg.4) that give different definitions for Hermitian inner product in $\mathbb{F}_4$.
The first paper (which has 380+ cites) gives the following:
$$ (\boldsymbol u, \boldsymbol v)_H = u_1v^2_1 + u_2v^2_2 + \ldots +
u_nv^2_n, $$ where $\boldsymbol u = \begin{pmatrix}u_1 & u_2 & \ldots
& u_n\end{pmatrix}$ and $\boldsymbol v = \begin{pmatrix}v_1 & v_2 &
\ldots & v_n\end{pmatrix}$.
The takeaway is that the second parameter (i.e., $\boldsymbol v$) is conjugated and $\boldsymbol u$ is left the same.
The second paper (which has 90 cites) gives the following definition:
The Hermitian inner product of two elements $a, b \in \mathbb{F}_4$ is
defined as $\langle a, b\rangle = a^\dagger b \in \mathbb{F}_4$ where
where ${}^{\dagger}$ denotes conjugation.
In other words, $a^\dagger = a^2$. The takeaway is that the first argument is conjugated, not the second.
If we try $\langle \bar{\omega}, \omega\rangle$ using the first definition (assuming $u = \begin{pmatrix}\bar{\omega}\end{pmatrix}$ and $\boldsymbol v = \begin{pmatrix}\omega\end{pmatrix}$):
$$\langle \bar{\omega}, \omega\rangle = \bar{\omega}\omega^2 = \bar{\omega}\bar{\omega}=\omega.$$
Using the second definition, we get:
$$\langle \bar{\omega}, \omega\rangle = \bar{\omega}^2\omega = \omega\omega=\bar{\omega}$$
Why are the two definitions different? Are they even describing the same inner product? Which definition should I follow and why not the other?
Best Answer
First of all, please note that the number of citations is not a measure of how correct a paper is.
That said, both definitions are "correct" and they describe the same inner product up to notational differences, but they follow different conventions. In a general mathematical context, it is more common to define Hermitian forms $\langle \cdot, \cdot \rangle$ to be linear in the first argument. In a physics context (especially where quantum mechanics is involved), it is more common to follow the opposite convention whereby $\langle \cdot, \cdot \rangle$ is linear in the second argument (in analogy to "bra-ket" notation).
An analogous conflict occurs with the usage of spherical coordinates: in different subfields of mathematics and the sciences, different conventions are used. All of these are "correct", but in either case the convention that one follows should be made explicit.
I would say that you should follow whichever convention you are more comfortable with. If you are not significantly more comfortable with one than the other, then you could also choose the convention that is more common in the relevant subfield. For instance, I would follow the second convention in the context of quantum-mechanics related fields (such as quantum information theory) and follow the first convention otherwise.