Post #4776256 · Elektrine

2026-09-19 09:44 UTC

Every nonzero vector is basically just an arrow: sure, they can have different lengths and point in different directions, but they all look like arrows. For bivectors the classification is different. Some look like flat surfaces: these are the ones you can write as u ∧ v for two vectors u and v. Some can't be written this way, but can be written as a sum of two terms u ∧ v + w ∧ z for some vectors u,v,w,z. In 3d or 4d space that's as bad as it gets. But in higher dimensions there are bivectors that can only be written as a sum of three terms, or four, etc. When space has dimension at most 2n, you need at most n terms. For trivectors the classification gets a lot more interesting. You can write every trivector as u ∧ v ∧ w when the dimension of space is low enough, namely 5-dimensional or lower. When you hit 6 dimensions you also get trivectors that you can only write as a sum of two terms, like u ∧ v ∧ w + a ∧ b ∧ c. The same holds in dimensions 7 and 8. But in 9 dimensional space, ALL HELL BREAKS LOOSE! There are *infinitely* many kinds of trivectors. 🌩️ To be honest: these new ones can all be written as a sum of three terms, like u ∧ v ∧ w + a ∧ b ∧ c + d ∧ e ∧ f. So you could say there's just one new kind. But that's not how mathematicians think about it. I really need to say more precisely what I mean by a 'kind' of trivector. [Typical way math professor starts to talk about what they really REALLY wanted to talk about.] (2/n)

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@johncarlosbaez@mathstodon.xyz 2026-09-19 10:08

Let's look at linear combinations of wedge products of k different vectors in n-dimensional space. We call these 'k-vectors'. The group GL(n) of linear transformations of n-dimensional space acts on the set of k-vectors. We can look at orbits of this group action. An orbit is what I'm calling a 'kind' of k-vector in n-dimensional space. There are finitely many kinds of k-vector in n-dimensional space in only these cases: k = 0 and n is anything. k = 1 and n is anything. k = 2 and n is anything. k = 3 and n < 9. k = 4 and n < 8. n-k = 4 and n < 8. n-k = 3 and n < 9. n-k = 2 and n is anything. n-k = 1 and n is anything. n-k = 0 and n is anything. You'll notice this list is palindromic! That's not a coincidence. I will just mutter two words for those in the know: Hodge duality. The case k = 3 and n = 9 is interesting. There are infinitely many kinds of trivectors in 9-dimensional space. There's a 3-dimensional space of kinds! It's connected to the Lie group E8, and other cool things! This is what I REALLY wanted to talk about, but I have to go shopping so for now I'll just point you to these: • Ernest B. Vinberg and Alexander G. Elashvili, A classification of the trivectors of a nine-dimensional space, Trudy Seminara po Vektornomu i Tenzornomu Analizu 18 (1978): 197–233. English version: Selecta Mathematica Sovietica 7, no. 1 (1988): 63–98. • Victor G. Kac, Some remarks on nilpotent orbits,Journal of Algebra 64, no. 1 (1980): 190–213. https://doi.org/10.1016/0021-8693(80)90141-6 These do the complex case. The real case is harder: • Mikhail Borovoi, Willem A. de Graaf, and Hông Vân Lê, Real graded Lie algebras, Galois cohomology, and classification of trivectors in ℝ⁹. https://arxiv.org/abs/2106.00246 (3/n)

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