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

TL;DR

This paper analyzes the structure of Adinkra gadgets derived from BC4 Coxeter group elements, revealing that most matrix elements are zero or take on only three specific values, forming a subspace akin to a tetrahedral molecule.

Contribution

It provides a comprehensive analysis of the Gadget matrix elements for all possible Adinkras with four colors and nodes, identifying the distribution of non-zero values and their geometric interpretation.

Findings

Out of over 1.3 billion matrix elements, only about 226 million are non-zero.

Non-zero elements take only three distinct values: -1/3, 1/3, and 1.

The non-zero structure forms a subspace similar to a body-centered tetrahedral molecule.

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 non-vanishing 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.