# Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately.

I have the definition

$$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots a_{n-1}b_1\dots b_p}\omega^{b_1\dots b_p}$$

where $\epsilon = \frac{1}{n!}\sqrt{|\det{g}|}\ dx^1\wedge\dots\wedge dx^n$ is the volume form on the manifold.

I thought I'd try to find a formula for $\star (dx^{\mu_1}\wedge\dots dx^{\mu_p})$ as an exercise. But I can't seem to get it to come out to a nice answer.

In particular I get lots of terms like

$$g^{\mu_{\alpha} b_{\alpha}}$$

from lowering indices. No texts that I've read seem to involve these! Could someone possibly tell me the correct formula for the result, and give me some hints as to how to do the calculation efficiently?

Many thanks!

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