arXiv subjects
Publications and source records attributed to Robert W. Donley Jr.
We survey the combinatorics of the Adinkra, a graphical device for solving differential equations in supersymmetry. These graphs represent an exceptional class of 1-factorizations with further augmentations. As a new feature, we characterize Adinkras using Latin rectangles.
We give formulas for enumerating directed paths in the graded poset of semi-magic squares of size three. We give two applications of these formulas: an advanced example of Vandermonde convolution for finite graded posets, and a direct method for deriving Regge symmetry formulas for un-normalized Clebsch-Gordan coefficients.
After reviewing the group structure and representation theory for the dihedral group $D_{2n},$ we consider an intertwining operator $Φ_ρ$ from the group algebra $\mathbb{C}[D_{2n}]$ into a corresponding space of semi-magic matrices. From this intertwining operator, one obtains the generating function for enumerating the associated semi-magic squares with fixed line sum and an algebra extending the circulant matrices. While this work complements the approach to $D_{2n}$ through permutation polytopes, we use only methods from representation theory.
We develop further properties of the matrices $M(m, n, k)$ defined by the author and W. G. Kim in a previous work. In particular, we continue an alternative approach to the theory of Clebsch-Gordan coefficients in terms of combinatorics and convex geometry. New features include a censorship rule for zeros, a sequence of 36-pointed stars of zeros, and another proof of Dixon's Identity. As a major application, we reinterpret the work of Raynal {\it et al.} on vanishing Clebsch-Gordan coefficients as a "middle-out" approach to computing $M(m, n, k).$