Eigenvalues and Critical Groups of Adinkras

TL;DR

This paper explores the spectral properties and critical groups of Adinkras, signed graphs used in supersymmetry, revealing their eigenvalue structure and introducing new methods for analyzing their algebraic invariants.

Contribution

It provides a detailed study of the eigenvalues of Adinkras' matrices and introduces a novel technique for analyzing their critical groups over polynomial rings.

Findings

Adinkras have exactly two distinct eigenvalues in their adjacency and Laplacian matrices.

The critical groups of Adinkras are characterized, especially their odd components.

A new method over polynomial rings is developed for analyzing critical groups.

Abstract

Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers critical groups over polynomial rings.