Submultiplets

Submultiplets are subsets of states within a larger multiplet that transform among themselves under a smaller symmetry group. In representation theory a multiplet is an irreducible representation of a group G. When the symmetry is reduced to a subgroup H, the G-representation may decompose, upon restriction to H, into a direct sum of irreducible H-representations. Each irreducible component is referred to as a submultiplet.

In physics, submultiplets help organize spectra when a high-symmetry system is observed or approximated under a lower symmetry. They explain why a single high-symmetry multiplet can contain several distinct families of states that behave as separate, smaller multiplets under the subalgebra or subgroup.

A standard illustration comes from flavor SU(3) in particle physics. The baryon octet forms an eight-dimensional representation of SU(3). When viewed through the SU(2) isospin subgroup, this octet decomposes into submultiplets with isospin I = 1/2 (two distinct doublets, containing the nucleons N and the Xi), I = 1 (the Sigma triplet), and I = 0 (the Lambda singlet). Similarly, meson octets decompose into isospin submultiplets consisting of pions (I = 1), kaons (I = 1/2), and an isospin I = 0 component.

In atomic and nuclear physics, submultiplets arise when rotational or other continuous symmetries are broken by external fields or interactions. For example, a total angular momentum J multiplet under SO(3) can split in a magnetic field, with states grouped into submultiplets labeled by the remaining symmetry such as M_J under SO(2).

Overall, submultiplets organize the structure of states under symmetry breaking or subgroups, clarifying degeneracies, selection rules, and transition patterns.