Adinkra (In)Equivalence From Coxeter Group Representations: A Case Study

TL;DR

This paper analyzes the 384-dimensional solution space of signed permutation matrices related to supersymmetry representations, proposing a new classification method using permutation group equivalence classes and a dual operator.

Contribution

It introduces an alternative classification of adinkra representations using permutation group S4 equivalence classes and a dual operator, expanding the understanding of supersymmetry algebra representations.

Findings

Partition of representations into three classes by the dual operator

Use of permutation group S4 for classification

Numerical analysis of signed permutation matrices

Abstract

Using a Mathematica code, we present a straightforward numerical analysis of the 384-dimensional solution space of signed permutation 4x4 matrices, which in sets of four provide representations of the GR(4,4) algebra, closely related to the N=1 (simple) supersymmetry algebra in 4-dimensional spacetime. Following after ideas discussed in previous papers about automorphisms and classification of adinkras and corresponding supermultiplets, we make a new and alternative proposal to use equivalence classes of the (unsigned) permutation group S4 to define distinct representations of higher dimensional spin bundles within the context of adinkras. For this purpose, the definition of a dual operator akin to the well-known Hodge star is found to partition the space of these GR(4,4) representations into three suggestive classes.