John Carlos Baez on Nostr: Vinberg and Elashvili classified all kinds of trivectors in ℂ⁹. But they looked ...

Vinberg and Elashvili classified all kinds of trivectors in ℂ⁹. But they looked at the space of trivectors modulo SL(n) rather than GL(n). Here SL(n) is the group of n×n matrices with determinant 1. So, they didn't count two trivectors as the same when they differ by an overall scale factor. This actually makes things more pretty, though it means the space has dimension 4 rather than 3.

They showed that up to the action of SL(n), every trivector is a linear combination of these 4 trivectors:

h₁ = x₁∧x₂∧x₃ + x₄∧x₅∧x₆ + x₇∧x₈∧x₉
h₂ = x₁∧x₄∧x₇ + x₂∧x₅∧x₈ + x₃∧x₆∧x₉
h₃ = x₁∧x₅∧x₉ + x₂∧x₆∧x₇ + x₃∧x₄∧x₈
h₄ = x₁∧x₆∧x₈ + x₂∧x₄∧x₉ + x₃∧x₅∧x₇

This looks very scary - almost random! But look at the first term in each sum:

x₁∧x₂∧x₃
x₁∧x₄∧x₇
x₁∧x₅∧x₉
x₁∧x₆∧x₈

Compare it to the chart below. In this chart, I've drawn a 3×3 grid, and all lines through the upper left-hand corner, where the purple line "wraps around". Each line has three points on it! And these points are

1 2 3
1 4 7
1 5 9
1 6 8

matching the pattern we just saw!

But that only explains the first term in each trivector h₁,h₂,h₃,h₄. Can you guess the pattern that explains other two terms in each trivector? I'll show you.

(4/n)