Adinkras

Adinkras are visual symbols ("originally created by the Akan of Ghana and the Gyaman of Cote d'Ivoire in West Africa" - wikipedia) representing different concepts. In the paper by Faux and Gates, the term "Adinkra" was used to refer to an edge-colored bipartite graph with height assignments on the vertices which contains all the data of a certain type of representation of the supersymmetry algebra. They look like this:



In classifying the off-shell representations of the supersymmetry algebra, Adinkras are a key tool. I have spent the last few days applying the new Google-funded code to classify (undashed) Adinkras, for a paper in preparation. On the first day, other than writing up a rather naive implementation of a classification algorithm, I spent most of my time hunting for this bug. The second day, today, was spent optimizing the algorithm (from 173.54 secs to 88.74 secs for all N=5 Adinkras).

Here is how to compute the canonical label and automorphism group of an undashed Adinkra. An isomorphism of Adinkras is a map of the vertices which leaves the heights unchanged, which is a graph isomorphism on the underlying graph with the option of permuting the colors which label the edges. Thus we form a partition of the vertices by height. Next, an edge u,v with color c becomes a triple of edges. One new vertex is inserted to represent the edge, and two edges come from this vertex to the vertices incident with the original edge. The third edge goes from a vertex representing the original edge to a vertex representing the color c. The partition then gets a cell consisting of the edges, and a cell consisting of the colors. Thus the edges may permute amongst themselves and the colors may permute amongst themselves.