Decomposing Grassmann Monomials for Superfield Expansions

Abstract

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 ${\Lambda }^n({\mathbb{C}}^{d_S}\otimes {\mathbb{C}}^{d_F})$, 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.

22 pages, 4 tables, Julia script attached