Many Correlation Tables are Molien Sequences

TL;DR

The paper introduces a new method for calculating rotational correlation tables using Molien sequences, extending to unitary groups and revealing implications for degeneracy observables in quantum systems.

Contribution

It presents a novel approach leveraging the Jordan-Schwinger form to connect correlation tables with Molien sequences, enhancing computational efficiency and broadening applicability to unitary symmetries.

Findings

Correlation tables can be represented as Molien sequences.

Extension to U(n) groups introduces new degeneracy observables.

Provides a new computational framework for symmetry analysis.

Abstract

Using the Jordan-Schwinger form of the quantum angular momentum eigenstates, it is straight-forward to define rotational correlation tables such that the columns are Molien sequences for finite rotational subgroup $G$. This realization gives a new and better means of calculation. Generalization to unitary symmetry $U(n)$ implies many more sequences, which determine degeneracy observables in the context of electronic, vibrational, and rotational motion. This leads us to discuss one physical significance of the Hilbert finite basis theorem.