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.
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BibTeX
@misc{ma2026lowtwist,
title = {{Low-Twist Matrix Covariants of Exterior Powers: Vanishing and Modular Phenomena}},
author = {Jingchuan Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25029},
eprint = {2609.25029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25029},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Low-Twist Matrix Covariants of Exterior Powers: Vanishing and Modular Phenomena
AU - Jingchuan Ma
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.25029
UR - https://arxiv.org/abs/2609.25029
AB - 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.
ER -