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

Abstract

We study morphisms from symmetric powers of exterior powers to determinant-twisted endomorphism representations of general linear groups. At the minimal positive determinant twist, we prove vanishing over every field of characteristic different from two: if the exterior degree r>= 3 is odd, the symmetric degree satisfies d>= 3, and the underlying space has dimension rd, then the corresponding equivariant Hom space is zero. The proof uses a block-exchange sign and a universal root-subgroup identity, so it also applies in small odd characteristics without semisimplicity. We then determine the second-twist spaces for trivectors in characteristic zero. In dimension 3m and degree 2m, they are scalar and one-dimensional for m = 2, and zero for m>= 3. Plethystic conjugation reduces the latter vanishing to an elementary weight-support bound for exterior powers of the ten-dimensional space of ternary cubics. Finally, over fields of characteristic zero or odd characteristic, an exact computer-assisted classification in dimension nine gives a one-dimensional scalar Hom space in characteristic five and zero in characteristic zero and in every odd characteristic other than five.