Irreducible Decomposition of Products of 10D Chiral Sigma Matrices

TL;DR

This paper introduces a symbolic computational method for decomposing products of 10D chiral sigma matrices into irreducible components, enabling efficient derivation of identities beyond trace calculations.

Contribution

The authors developed a Mathematica program that automates the irreducible decomposition of 10D chiral sigma matrix products, facilitating advanced algebraic manipulations.

Findings

Successfully derived new identities involving sigma matrices

Provided a publicly available computational tool

Streamlined complex algebraic calculations in high-dimensional spinor algebra

Abstract

We review the enveloping algebra of the 10 dimensional chiral sigma matrices. To facilitate the computation of the product of several chiral sigma matrices we have developed a symbolic program. Using this program one can reduce the multiplication of the sigma matrices down to linear combinations of irreducilbe elements. We are able to quickly derive several identities that are not restricted to traces. A copy of the program written in the Mathematica language is provided for the community.