[Paper Review] Adinkra Isomorphisms and `Seeing' Shapes with Eigenvalues

Keith Burghardt, S. James Gates|arXiv (Cornell University)|Dec 12, 2012

Advanced Numerical Analysis Techniques5 references3 citations

TL;DR

This paper presents a novel, efficient algorithm to determine shape isomorphism between adinkras—graphical representations of one- and two-dimensional supersymmetry algebras—by computing eigenvalues of a total permutation matrix derived from color matrices. The method uniquely identifies isomorphic adinkras regardless of node labeling or ordering, offering a computationally friendly alternative to prior methods.

ABSTRACT

We create an algorithm to determine whether any two graphical representations (adinkras) of equations possessing the property of supersymmetry in one or two dimensions are isomorphic in shape. The algorithm is based on the determinant of `permutation matrices' that are defined in this work and derivable for any adinkra.

Motivation & Objective

• To develop a computationally efficient and reliable algorithm for determining whether two adinkras are isomorphic in shape, regardless of node labeling or ordering.
• To overcome the limitations of prior algorithms that were computationally inefficient and required complex node reordering and height determination.
• To provide a method that is both mathematically rigorous and suitable for implementation on parallel computing architectures.
• To establish a systematic way to compare adinkras by leveraging spectral invariants (eigenvalues) of a derived total permutation matrix.
• To demonstrate the method's superiority over existing approaches through direct comparison on non-trivial adinkra examples.

Proposed method

• Construct color matrices for each adinkra color (e.g., green, yellow, red) based on edge directions and node positions, assigning parameters β_I or β_I^(-1) depending on relative node height.
• Form a total permutation matrix by multiplying the color matrices in a specific order, representing the full structure of the adinkra.
• Compute the eigenvalues of the absolute value of the total permutation matrix, which serve as a spectral invariant for shape comparison.
• Compare the eigenvalue spectra of two adinkras: identical spectra imply isomorphism, while differences confirm non-isomorphism.
• Use the trace of the matrix with all parameters set to 1 to recover chromocharacters, which detect line-dashing differences (twisted vs. untwisted multiplets).
• Leverage the fact that eigenvalues are invariant under permutation of node labels, making the method robust to arbitrary labeling.

Experimental results

Research questions

• RQ1 Can a spectral invariant derived from permutation matrices reliably determine isomorphism between two adinkras without relying on node reordering?
• RQ2 How does the eigenvalue spectrum of the total permutation matrix relate to the underlying shape of the adinkra?
• RQ3 Can this method efficiently distinguish non-isomorphic adinkras that are difficult to compare via visual or conventional algebraic means?
• RQ4 To what extent does the method preserve information about line dashing (chromocharacters) while focusing on shape isomorphism?
• RQ5 How does this algorithm compare in efficiency and accuracy to previous isomorphism detection methods for adinkras?

Key findings

• The eigenvalue spectrum of the total permutation matrix uniquely characterizes the shape of an adinkra, enabling reliable isomorphism detection.
• The method correctly identifies that the two adinkras in Figure 6 are not isomorphic, as their eigenvalue spectra differ, despite both being (6|8|2) adinkras.
• The algorithm avoids the need for node reordering and height determination, which were required in earlier approaches and significantly increased computational cost.
• The chromocharacter, recoverable as the trace when all parameters are set to 1, correctly identifies differences in line dashing, confirming the method's ability to detect twisted multiplets.
• The algorithm is computationally efficient and parallelizable, offering a significant improvement over prior methods in terms of implementation and scalability.
• The method successfully distinguishes non-isomorphic adinkras such as the bow tie and diamond, even when they share the same number of nodes and edges, by detecting differences in their eigenvalue spectra.