In n dimensions, the space of trivectors has dimension n choose 3, since you can find a basis of trivectors like xᵢ∧ xⱼ∧xₖ but we have xᵢ∧xⱼ=−xⱼ∧xᵢ .
On the other hand, the group GL(n) of all invertible n×n matrices has dimension n².
The space of 'kinds' of trivectors is the space of trivectors modulo the action of GL(n). Thus, we expect its dimension to be at most n choose 3 minus n². This first becomes positive when n hits 9.
For n less than n, the space of kinds of trivectors is 0-dimensional - in fact a finite set. But when n = 9, it's 3-dimensional, as we might guess from
84 - 81 = 3
(This is not a proof, just a quick guess. But it's correct.)
(4/n)
