Adinkras From Ordered Quartets of BC${}_4$ Coxeter Group Elements and Regarding 1,358,954,496 Matrix Elements of the Gadget

Abstract

We examine values of the Adinkra Holoraumy-induced Gadget representation space metric over all possible four-color, four-open node, and four-closed node adinkras. Of the 1,358,954,496 gadget matrix elements, only 226,492,416 are nonvanishing and take on one of three values: −1/3, 1/3, or 1 and thus a subspace isomorphic to a description of a body-centered tetrahedral molecule emerges.

Spacetime Spin and Chirality Operators for Minimal 4D, $\cal N$ = 1 Supermultiplets From BC${}_4$ Adinkra-Tessellation of Riemann Surfaces

“…We emphasize these are conjectures as we expect these to hold over all the adinkras associated with BC 4 . The work in [5] shows this number is 36,864 adinkras.…”

“…Here we use the "read down" convention for relating cycle notation to matrix notation of permutations as discussed in the work of[5].…”