I have been reading Introduction to Smooth Manifolds by John M.Lee
The basis for the space of alternating covariant k-tensors $\Lambda^k(V^*)$, is given by the collection
{ $\epsilon^I$; $\mathit{I}$ is an increasing multi-index of length k }
My question is why $\mathit{I}$ has to been an increasing multi-index?
Can someone show me why $\mathit{I}$ cannot be any multi-index of length k?
Best Answer
All the indices have to be different (because it's alternating), and if two multiindices are permutations of each other then the corresponding vectors are the same up to a sign.
Demanding that the indices are increasing is a way to force them all to be different while simultaneously picking a single element out of all the equivalent permutations.