Computing SL(2,C) Central Functions with Spin Networks

TL;DR

This paper introduces a combinatorial approach using trace diagrams to compute SL(2,C) central functions and provides two algorithms for their calculation.

Contribution

It presents a novel graph-based method and algorithms for efficiently computing SL(2,C) central functions from trace diagrams.

Findings

Trace diagrams provide a combinatorial description of central functions.

Two Mathematica algorithms enable practical computation of these functions.

The methods facilitate analysis of free group representations into SL(2,C).

Abstract

Let G=SL(2,C) and F_r be a rank r free group. Given an admissible weight in N^{3r-3}, there exists a class function defined on Hom(F_r,G) called a central function. We show that these functions admit a combinatorial description in terms of graphs called trace diagrams. We then describe two algorithms (implemented in Mathematica) to compute these functions.