Adinkras for Mathematicians
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 6, June 2014, Pages 3325–3355 S 0002-9947(2014)06031-5 Article electronically published on February 6, 2014 ADINKRAS FOR MATHEMATICIANS YAN X. ZHANG Abstract. Adinkras are graphical tools created to study representations of supersymmetry algebras. Besides having inherent interest for physicists, the study of adinkras has already shown non-trivial connections with coding the- ory and Clifford algebras. Furthermore, adinkras offer many easy-to-state and accessible mathematical problems of algebraic, combinatorial, and computa- tional nature. We survey these topics for a mathematical audience, make new connections to other areas (homological algebra and poset theory), and solve some of these said problems, including the enumeration of all hyper- cube adinkras up through dimension 5 and the enumeration of odd dashings of adinkras for any dimension. 1. Introduction In a series of papers starting with [11], different subsets of the “DFGHILM collaboration” (Doran, Faux, Gates, H¨ubsch, Iga, Landweber, Miller) have built and extended the machinery of adinkras. Following the ubiquitous spirit of visual diagrams in physics, adinkras are combinatorial objects that encode information about the representation theory of supersymmetry algebras. Adinkras have many intricate links with other fields such as graph theory, Clifford theory, and coding theory. Each of these connections provide many problems that can be compactly communicated to a (non-specialist) mathematician. This paper is a humble attempt to bridge the language gap and generate communication. In short, adinkras are chromotopologies (a class of edge-colored bipartite graphs) with two additional structures, a dashing condition on the edges and a ranking condition on the vertices.
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