A proposal on culling & filtering a coxeter group for 4D, N $$ \mathcal{N} $$ = 1 spacetime SUSY representations: revised

Abstract

We present an expanded and detailed discussion of 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, N = 1 spacetime supersymmetry algebra.

Adinkras from ordered quartets of BC4 Coxeter group elements and regarding 1,358,954,496 matrix elements of the Gadget

“…These results support the concept of "SUSY holography" by showing the "angles" [38][39][40][41][42] between any two adinkras constructed from ordered quartets of BC 4 elements take on exactly and only the same values as the "angles" [40] between any of the 4D, N = 1 supermultiplets with minimal numbers of bosons and fermions. The presentation in section 3 is made in terms of a visual/graphical representation.…”

“…In the work of [42], there was presented and released a list of the values of the and parameters, though the calculations of these occurred contemporaneously with work of [44]. On the basis of this "library" (that we will subsequently call the "small BC 4 library," algorithms and codes, to be discussed later, were written in order to calculate the values of the quadratic forms (2.8) on the and adinkra parameter spaces.…”

“…Section 6 goes to the consideration of looking at the effects of considering the role of "twisting" quartets by introducing relative sign factors among the components within quartets of the "small BC 4 library" that results strictly from taking a single representation of the elements of BC 4 and using them to construct supersymmetric quartets that satisfy the "Garden Algebra" conditions. This material has exactly appeared previously in [42] and the reader familiar with this discussion can skip this.…”

“…In facing the problem of reconstructing 4D simple supermultiplets from adinkras, unlike the successful paths shown for the analogous problem in 2D, another path has been pursued also [36][37][38][39][40][41][42]. This alternate path is based on the fact that closed four-cycles for four color valise adinkras possess naturally an SO(4) symmetry.…”

“…Also as was explicitly discussed [42], given a complete BC 4 description of an adinkra, one can construct "complements" of any adinkra. These also include "anti-podal" representations where one representation can be obtained from the other by simply re-defining either all of closed nodes (or open nodes) by a minus sign.…”