An $N$-independent tensor decomposition for SU($N$)

TL;DR

This paper introduces an N-independent tensor decomposition for SU(N) groups, enabling a unified treatment of color structures in QCD across different N values using Young diagrams.

Contribution

It develops a generalized N-independent reduction of SU(N) tensor products by extending the Littlewood-Richardson rule to pairs of Young diagrams, applicable to quarks and antiquarks.

Findings

Provides a unified framework for SU(N) tensor products

Enables systematic decomposition of color structures in QCD

Generalizes the Littlewood-Richardson rule for pairs of Young diagrams

Abstract

To facilitate a simultaneous treatment of an arbitrary number of colors in representation theory-based descriptions of QCD color structure, we derive an $N$-independent reduction of SU($N$) tensor products. To this end, we label each irreducible representation by a pair of Young diagrams, with parts acting on quarks and antiquarks. By combining this with a column-wise multiplication of Young diagrams, we generalize the Littlewood-Richardson rule for the product of two Young diagrams to the product of two Young diagram pairs, achieving a general-$N$ decomposition.