Compact construction algorithms for the singlets of SU(N) over mixed tensor product spaces

TL;DR

This paper introduces an efficient algorithm for constructing projection operators onto singlet irreducible representations of SU(N) over mixed tensor product spaces, improving computational efficiency for these specific cases.

Contribution

The paper presents a new, efficient algorithm for constructing projection operators onto SU(N) singlets in mixed Fock space components, avoiding complex translation steps.

Findings

The algorithm simplifies the construction of singlet projections.

It provides transparent access to N dependence in the representations.

Comparison shows advantages over previous methods.

Abstract

The irreducible representations of SU(N) over a mixed quark-antiquark Fock space component have been studied for many years. In analogy to the case for the quark-only Fock space component, there exist efficient tools to classify the irreducible representations of SU(N) over a mixed tensor product space using tableaux. Unlike the quark-only case, the only general algorithm known to us for constructing the associated projection operators onto irreducible multiplets involves translating N-1 fundamental factors into an antifundamental factor using the Leibniz rule, which turns out to be computationally extremely inefficient. If one is interested only in singlets, this problem can be entirely avoided as is demonstrated below where we provide an efficient algorithm to construct the projection operatators onto the irreducible representations of dimension 1 of the special unitary group SU(N) over a mixed Fock space component that transparently gives access to N dependence, and discuss the relative merits in comparison to an alternative algorithm briefly discussed in a different context by Keppeler and Sjoedahl.