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

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.

23 pages, 6 figures (v.2: numerical data corrections in appendix C)

References in corpus (2)

• The Real Anatomy of Complex Linear Superfields
• Golden Ratio Controlled Chaos in Supersymmetric Dynamics

Cited by in corpus (15)

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• The 300 "Correlators" Suggests 4D, $\mathcal{N}$ = 1 SUSY Is a Solution to a Set of Sudoku Puzzles
• Is It Possible To Embed A 4D, N = 4 Supersymmetric Vector Multiplet Within A Completely Off-Shell Adinkra Hologram?
• Generating all 36,864 Four-Color Adinkras via Signed Permutations and Organizing into $\ell$- and $\tilde{\ell }$-Equivalence Classes
• Adinkras From Ordered Quartets of BC${}_4$ Coxeter Group Elements and Regarding 1,358,954,496 Matrix Elements of the Gadget
• Think Different: Applying the Old Macintosh Mantra to the Computability of the SUSY Auxiliary Field Problem
• Adinkra Height Yielding Matrix Numbers: Eigenvalue Equivalence Classes for Minimal Four-Color Adinkras
• Adinkras From Ordered Quartets of BC4 Coxeter Group Elements and Regarding Another Gadget's 1,358,954,496 Matrix Elements
• A Proposal On Culling & Filtering A Coxeter Group For 4D, N = 1 Spacetime SUSY Representations
• Properties of HYMNs in Examples of Four-Color, Five-Color, and Six-Color Adinkras
• Examples of 4D, N = 2 Holoraumy
• On the Ubiquity Of Electromagnetic-Duality Rotations in 4D, N = 1 Holoraumy Tensors for On-Shell 4D Supermultiplets
• Infinite-Dimensional Algebraic $\mathfrak{Spin}$($N$) Structure in Extended/Higher Dimensional SUSY Holoraumy for Valise and On-Shell Supermultiplet Representations