# Efficient algorithm for representations of U(3) in U(N)

#### Abstract

An efficient algorithm for enumerating representations of U(3) that occur in a representation of the unitary group U(N) is introduced. The algorithm is applicable to U(N) representations associated with a system of identical fermions (protons, neutrons, electrons, etc.) distributed among the N=(η+1)(η+2)∕2 degenerate eigenstates of the ηth level of the three-dimensional harmonic oscillator. A C++ implementation of the algorithm is provided and its performance is evaluated. The implementation can employ OpenMP threading for use in parallel applications. Program summary: Program Title: UNtoU3.h Program files doi: http://dx.doi.org/10.17632/3g4w8f9vdk.1 Licensing provisions: MIT Programming language: C++ Nature of problem: The determination of the complete set of U(3) irreducible representations (irreps) that occurs in a representation of U(N), where N=(η+1)(η+2)∕2 is the degeneracy of the ηth harmonic oscillator shell. Solution method: The resulting set of U(3) irreps is determined by applying a simple difference relation to the U(3) weight distribution of the Gelfand basis states spanning a given U(N) irrep.

*This paper has been withdrawn.*