Adinkras From Ordered Quartets of BC4 Coxeter Group Elements and Regarding Another Gadget's 1,358,954,496 Matrix Elements

TL;DR

This paper introduces a second scalar Gadget for adinkra graphs, computes its extensive matrix elements over BC4 related adinkras, and explores their automorphisms, advancing understanding of adinkra symmetries.

Contribution

It verifies the existence of a second Gadget for four-color adinkras and calculates its large matrix representation using analytical and computational methods.

Findings

Confirmed a second Gadget exists for four-color adinkras.

Computed 1,358,954,496 matrix elements of this Gadget.

Analyzed automorphisms of BC4 related adinkra graphs.

Abstract

A Gadget, more precisely a scalar Gadget, is defined as a mathematical calculation acting over a domain of one or more adinkra graphs and whose range is a real number. A 2010 work on the subject of automorphisms of adinkra graphs, implied the existence of multiple numbers of Gadgets depending on the number of colors under consideration. For four colors, this number is two. In this work, we verify the existence of a second such Gadget and calculate (both analytically and via explicit computer-enabled algorithms) its 1,358,954,496 matrix elements over 36,864 minimal valise adinkras related to the Coxeter Group BC4.