Abstract
We review the mathematical tools required to cull and filter representations of the Coxeter Group $BC_4$ into providing bases for the construction of minimal off-shell representations of the 4D, $ {\cal N}$ = 1 spacetime supersymmetry algebra. Of necessity this includes a description of the mathematical mechanism by which four dimensional Lorentz symmetry appears as an emergent symmetry in the context of one dimensional adinkras with four colors described by the Coxeter Group $BC_4$.
Highlights
In the second section, we review results that are standard to the usual Dirac matrices appropriate for a four dimensional Minkowski space
We present an expanded and detailed discussion of the mathematical tools required to cull and filter representations of the Coxeter Group BC4 into providing bases for the construction of minimal off-shell representations of the 4D, N = 1 spacetime supersymmetry algebra
We review results that are standard to the usual Dirac matrices appropriate for a four dimensional Minkowski space
Summary
The label (R) written in (4.2) takes its values over these representations and a more detailed description is given later This startlingly smaller number is mostly determined by the permutation elements from which any L-matrix is constructed. Given two quartets ( L(IR) )i ˆ and ( L(IR ) )i ˆ where the elements in the second set are related to the first by replacing at least one Boolean factor by its antonym, there exist a 4 × 4 Boolean factor matrix denoted by [A(R, R )] I J which acts on the color space of the links We have enunciated the rich mathematical structure imposed on the Coxeter Group BC4 when analyzed through the lens of the “Garden Algebra” GR(4,4)
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