# Clarifying the definition of Wedge Product — a detail about factorial prefactor

Munkres book on Manifolds constructs a wedge product by defining the following sum on $$f$$ (an alternating $$k$$-tensor on $$V$$) and $$g$$ (an alternating $$l$$-tensor on $$V$$): $$(f \wedge g)(v_1,...,v_{k+l}) = \frac{1}{k!l!}\sum_{\sigma} \text{sgn }\sigma f(v_{\sigma(1)},...,v_{\sigma(k)})g(v_{\sigma(k+1)},...v_{\sigma(k+l)})$$

Here $$\sigma$$ is a permutation on $$k+l$$ distinct elements.

He goes on to justify why there is the factor $$\frac{1}{k!l!}$$. I see that there are some permutations that will permute the first set of vectors $$\{v_1,...v_k\}$$ and the second set of vectors $$\{v_{l+1},...v_{l+k}\}$$ amongst themselves. I don't understand what will happen with the sign though. Munkres says:

because $$f$$ and $$g$$ are alternating tensors, the values of $$f$$ and $$g$$ change by being multiplied by the same sign

I'm having trouble seeing this. Knowing whether the permutation on $$f$$ is even or odd doesn't determine the evenness/oddness of $$g$$ does it?

The point is this: If you have a permutation $$\sigma = \tau\times\pi$$, where $$\tau$$ permutes $$1,\dots,k$$ and $$\pi$$ permutes $$k+1,\dots,k+\ell$$, then $$\text{sgn}\,\sigma = (\text{sgn}\,\tau)(\text{sgn}\,\pi)$$, and \begin{align*} \text{sgn}\,\sigma f(&v_{\sigma(1)},\dots,v_{\sigma(k)})g(v_{\sigma(k+1)},\dots,v_{\sigma(k+\ell)}) \\ &=\text{sgn}\,\sigma f(v_{\tau(1)},\dots,v_{\tau(k)})g(v_{\pi(k+1)},\dots,v_{\pi(k+\ell)}) \\ &= \big(\text{sgn}\,\sigma\,\text{sgn}\,\tau\,\text{sgn}\,\pi\big)f(v_1,\dots,v_k)g(v_{k+1},\dots,v_{k+\ell})\\ &= f(v_1,\dots,v_k)g(v_{k+1},\dots,v_{k+\ell}). \end{align*} And of course there are $$k!\ell!$$ such terms.
• There is an alternate definition of wedge product which involves summing over only the $(k, l)$-shuffling permutations, instead of all $(k+l)$-permutations. This means that we no longer require the $1\over k!l!$ factor. But I don't quite have the proof of the equivalence of these two definitions. I am not sure if this is the right place to bring this up, but I asked a question about this a couple of months ago and it did not receive any attention. It would be great if you could look at it. Thanks. – feynhat Jul 1 at 17:28