外幂的低扭转矩阵协变：消失与模现象
Low-Twist Matrix Covariants of Exterior Powers: Vanishing and Modular Phenomena
- Fuzhou University Zhicheng College（福州大学至诚学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究外幂表示间的GL等变态射，在特征零确定第二扭转空间，并在奇特征证明最小扭转Hom空间消失，给出计算机辅助的孤立标量类。
AI中文摘要：
我们研究多项式表示 $\operatorname{Sym}^d(\Lambda^r V)$ 与 $\operatorname{End}(V)\otimes\det(V)^\ell$ 之间的 $GL(V)$-等变态射。标量矩阵施加必要条件 $rd=n\ell$，这挑出了前两个行列式扭转。在特征零情形，我们确定了移动族 $d=2m$ 与 $n=3m$ 中三向量的第二扭转空间。等价地，对每个 $m\geq 2$，我们证明 $\langle h_{2m}[e_3],s_{(3,2^{3m-2},1)}\rangle=0$，而 $(2^{3m})$ 处的标量系数在 $m=2$ 时等于一，在 $m\geq 3$ 时等于零。证明将移动三列系数转化为三行外幂对称积，并将整个族约化为 $\operatorname{Sym}^3(k^3)$ 的两个有限外幂分解。对于最小扭转，我们在每个特征不等于二的域上给出一个实际 Hom 证明：若 $r\geq 3$ 为奇数，$d\geq 3$，且 $n=rd$，则相应的 Hom 空间消失。此证明使用块交换符号和泛根子群，因此在非半单的奇特征情形下仍然有效。一个精确的计算机辅助附录记录了 $\operatorname{Sym}^6(\Lambda^3 k^9)$ 的一个孤立特征五标量类，并解释其除幂对偶泛函为何在所有纯六次幂上消失。
英文摘要：
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.
