I am figthting trying to convert the general form of a Yang Mills Lagrangian (or lagrangian density, better said), i.e.,
$ \mathcal{L}_{YM} = \frac{1}{2}\text{tr}\left(F \wedge \ast F\right)$
to the local coordinates expression, i.e.,
$ \mathcal{L}_{YM} = \frac{1}{4}\text{tr}\left(F_{\mu\nu}F^{\mu\nu}\right)\text{Vol}$
where "Vol" stands for the volume element $\sqrt{|{\text{det} g_{\mu\nu}}|}$.
The definition of the Hodge star operator I use:
$\alpha \wedge \ast\beta = \langle\alpha, \beta\rangle \text{Vol}$
I think my problem is in the inner product. While I know that the inner product of two 1-forms is $\langle\alpha,\beta\rangle = g^{\mu\nu}\alpha_{\mu}\beta_{\nu}$, for general p-forms (that is what I need because $F$ is a 2-form) my book says that if $e^{1}, e^{2}, ..., e^{p}$ and $f^{1}, ..., f^{p}$ are 1-forms, then their inner product is defined as:
$\langle e^{1}\wedge...\wedge e^{p}, f^{1}\wedge...\wedge f^{p}\rangle = \text{det}[\langle e^{i}, f^{j}\rangle]$
(right hand is the determinant of a $p \times p$ matrix of inner products.
The problem is that what I get is always zero when doing the determinant, there's something that I am missing, so I can't go from the first lagrangian (density) to the second one in local coordinates.
Thank you.
If $e^i$ is an orthonormal basis of $1$-forms (i.e. $\langle e^i, e^j \rangle = \delta^{ij}$), then the determinant formula gives
$$ \langle e^i \wedge e^j, e^k \wedge e^l \rangle = \delta^{ik}\delta^{jl} - \delta^{il}\delta^{jk}.$$
Thus if we have two-forms $F = F_{ij} e^i \wedge e^j$ and similarly $G$, bilinearity of the inner product gives $$\langle F,G \rangle = F_{ij} G_{kl}(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk})=F_{ij}G_{ij}-F_{ij}G_{ji}=2F_{ij}G_{ij} = 2F_{ij}G^{ij}.$$
where the second-last equality uses the antisymmetry of forms and the last one uses the fact that we were computing in an orthonormal basis. Now we're pretty much done: we have
$$ F^a \wedge \star F^a = \langle F^a, F^a\rangle \text{Vol} = 2F^a_{ij}F^{aij} \text{Vol}.$$
I'm not sure where the factor of $\frac14$ is coming from: maybe your source is using a different normalization for the wedge product, Hodge dual or inner product?