Low-Twist Matrix Covariants of Exterior Powers: Vanishing and Modular Phenomena

Abstract

We study GL(V)-equivariant morphisms between the polynomial representations Symᵈ(ΛʳV) and End(V) ⊗ det(V) to the ℓ-th tensor power. Scalar matrices impose the necessary condition rd = nℓ, which organizes the first two determinant twists studied here. In characteristic zero we determine the second-twist spaces for trivectors in the moving family d = 2m and n = 3m. Equivalently, for every m ≥ 2 we prove ⟨h₂ₘ[e₃], s₍₃,₂³ᵐ⁻²,₁₎⟩ = 0, while the scalar coefficient at (2³ᵐ) equals one for m = 2 and zero for m ≥ 3. The proof converts the moving three-column coefficient to a three-row exterior plethysm and reduces the entire family to two finite exterior-power decompositions of Sym³(k³). For the minimal twist we give an actual-Hom proof over every field of characteristic different from two: if r ≥ 3 is odd, d ≥ 3, and n = rd, then the corresponding Hom space vanishes. This proof uses block-swap signs and universal root subgroups, and therefore remains valid in nonsemisimple odd characteristic. An exact computer-assisted appendix records an isolated characteristic-five scalar class for Sym⁶(Λ³k⁹) and explains why its divided-dual functional vanishes on all pure sixth powers.