用于超场展开的格拉斯曼单项式分解
Decomposing Grassmann Monomials for Superfield Expansions
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中文总结 AI 辅助
本文提出一种通用表示论方法，用于分解超空间构建中超场展开所需的格拉斯曼单项式，通过Weyl特征比较计算分支重数，验证分解正确性并提供Julia脚本实现算法。
中文摘要 AI 辅助
超空间通过反对易的格拉斯曼坐标扩展时空，其指标可在自旋群与味群下变换。将格拉斯曼单项式分解为独立的不变结构是构建超场展开的核心步骤，但在扩展超空间中该过程难度日益增大。本文提出一种通用的表示论方法，用于分解空间Λⁿ(ℂᵈ_S⊗ℂᵈ_F)，其中d_S和d_F分别为自旋表示与味表示的维数。通过在确定性有理样本点处精确比较Weyl特征，计算GL(m)↓Sp(m)和GL(m)↓SO(m)的分支重数，避免使用Littlewood-Richardson修正规则。本文确立了特征假设的完备性并分析了计算复杂度，表明其复杂度为n的分拆数和群秩的多项式函数，不依赖Littlewood-Richardson组合学。通过维数检验验证了所得分解的正确性，提供了配套的独立Julia脚本，该脚本可实现任意秩群的算法并生成与不变张量的显式收缩。
英文摘要
Superspace extends spacetime by anticommuting Grassmann coordinates, whose indices may transform under spin and flavour groups. Decomposing Grassmann monomials into independent invariant structures is a central step in constructing superfield expansions, but becomes increasingly difficult in extended superspace. We present a general representation-theoretic procedure for decomposing the space $Λ^n(\mathbb{C}^{d_S}\otimes \mathbb{C}^{d_F})$, where $d_S$ and $d_F$ are the dimensions of the spin and flavour representations. The branching multiplicities for $\mathrm{GL}(m)\downarrow{\mathrm{Sp}(m)}$ and $\mathrm{GL}(m)\downarrow{\mathrm{SO}(m)}$ are computed by exact Weyl-character comparison at deterministic rational sample points, avoiding Littlewood-Richardson modification rules. We establish completeness of the character ansatz and analyse the computational complexity, showing that it is polynomial in the number of partitions of $n$ and in the group rank, without dependence on Littlewood-Richardson combinatorics. We validate the resulting decompositions via dimension checks. We provide an accompanying standalone Julia script which implements this algorithm for groups of arbitrary rank, and produces the explicit contractions with invariant tensors.
