Central Values for Clebsch-Gordan coefficients

TL;DR

This paper explores advanced properties of matrices related to Clebsch-Gordan coefficients, introducing combinatorial and geometric methods, and offers new proofs and interpretations for their computation and zero patterns.

Contribution

It presents new combinatorial and geometric insights into Clebsch-Gordan coefficients, including zero patterns, a new proof of Dixon's Identity, and a reinterpretation of existing computational methods.

Findings

Identification of a censorship rule for zeros in matrices

Introduction of a 36-pointed star zero pattern

A new proof of Dixon's Identity

Abstract

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 $M(m, n, k).$