# Adinkra Height Yielding Matrix Numbers: Eigenvalue Equivalence Classes for Minimal Four-Color Adinkras

## TL;DR

This paper introduces a comprehensive classification of eigenvalues, called Height Yielding Matrix Numbers (HYMNs), for all 36,864 minimal four-color adinkras, extending previous eigenvalue equivalence concepts to non-valise cases.

## Contribution

It extends eigenvalue equivalence classes to non-valise adinkras and provides a complete classification of node-lifting eigenvalues for all associated adinkras.

## Key findings

- Defined Height Yielding Matrix Numbers (HYMNs) for adinkras.
- Classified eigenvalues into HYMN equivalence classes.
- Developed Python code for eigenvalue classification.

## Abstract

An adinkra is a graph-theoretic representation of spacetime supersymmetry. Minimal four-color valise adinkras have been extensively studied due to their relations to minimal 4D, $\cal N$ = 1 supermultiplets. Valise adinkras, although an important subclass, do not encode all the information present when a 4D supermultiplet is reduced to 1D. Eigenvalue equivalence classes for valise adinkra matrices exist, known as $\chi_{\rm o}$ equivalence classes, where valise adinkras within the same $\chi_{\rm o}$ equivalence class are isomorphic in the sense that adinkras within a $\chi_{\rm o}$-equivalence class can be transformed into each other via field redefinitions of the nodes. We extend this to non-valise adinkras, via Python code, providing a complete eigenvalue classification of "node-lifting" for all 36,864 valise adinkras associated with the Coxeter group $BC{}_4$. We term the eigenvalues associated with these node-lifted adinkras Height Yielding Matrix Numbers (HYMNs) and introduce HYMN equivalence classes. These findings have been summarized in a $Mathematica$ notebook that can found at the HEPTHools Data Repository (https://hepthools.github.io/Data/) on GitHub.