Enumerative Gadget Phenomena for $(4,1)$-Adinkras

TL;DR

This paper investigates the combinatorial properties of adinkras related to supersymmetry, revealing symmetry phenomena through symbolic computation and group theory, and proposes generalizations for different parameters.

Contribution

It provides a symbolic computation of gadgets for (4,1)-adinkras, explains observed symmetry phenomena using group theory and combinatorics, and suggests potential generalizations.

Findings

Few gadget values appear, indicating high symmetry.

Group theory explains the observed phenomena.

Provides suggestions for generalizing gadgets to other parameters.

Abstract

Adinkras are combinatorial objects developed to study supersymmetry representations. Gates et al. introduced the "gadget" as a function of pairs of adinkras, obtaining some mysterious results for $(n=4, k=1)$ adinkras with computer-aided computation. Specifically, very few values of the gadget actually appear, suggesting a great deal of symmetry in these objects. In this paper, we compute gadgets symbolically and explain some of these observed phenomena with group theory and combinatorics. Guided by this work, we give some suggestions for generalizations of the gadget to other values of the $n$ and $k$ parameters.