John Carlos Baez on Nostr: Let's look at linear combinations of wedge products of k different vectors in ...

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 case:

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 n = 3 and n = 9 is interesting. There are infinitely many kinds of trivectors in 9-dimensional space. There's a 4-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

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