What is the constant C for Hodge dual in tensor notation?

[praharmitra]

So I know that the Hodge dual of a p-form $A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p}$ in d dimensions is given by

$$(*A)^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}} = C\epsilon^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}\mu_1 \mu_2 \cdot \cdot \cdot \mu_p}A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p}$$
where C is some number coefficient. I was wondering what the
constant C is for general p-forms in general d dimensions.
Also, what is the inverse relation? (I'm guessing it's the
same as above, but just checking.)

 

[Bill_K]

C can be anything you like, but if you use C = 1/p! where p is the number of contracted indices, the same formula works for both this formula and the inverse relation.