Paul Schwahn on Nostr: There actually is a trick in this particular case! Though it's a bit more complicated ...

There actually is a trick in this particular case! Though it's a bit more complicated because it uses spinors.

Let's interpret ℂ¹⁰ as the (complex) vector representation of the rotation group SO(10). This is also a representation of the spin group Spin(10), because Spin(10) double covers SO(10).

Next, take ℂ¹⁶ be either of the complex spinor representations of Spin(10) - the left-handed or the right-handed one, it doesn't matter.

Claim: There is a natural (i.e. Spin(10)-equivariant) isomorphism Λ³ℂ¹⁰ ≅ Λ²(ℂ¹⁶)*.

I'm not entirely sure what the isomorphism looks like - here's a guess: take a trivector 𝑥∧𝑦∧𝑧, and a pair of (left-handed) spinors ϕ,ψ. Then the Clifford product 𝑥⋅𝑦⋅𝑧⋅ϕ is a right-handed spinor, and we can pair it with ψ to obtain a complex number. My hope is that this is alternating in ϕ,ψ, and therefore defines an equivariant map Λ³ℂ¹⁰ → Λ²(ℂ¹⁶)*.