Branching rules for higher fundamental representations and arbitrary nilpotent types

Determine the branching of the fundamental representation L(ω_k) of sl_n to an sl_2-subalgebra of arbitrary partition type [d_1,d_2,…,d_m] for k≥3, including an explicit determination of its irreducible sl_2-multiplicities.

Background

The paper develops a recursion formula for multiplicities in the restriction of irreducible sl_n-representations to sl_2-subalgebras. To use this recursion, the restrictions of the fundamental representations L(ω_k) serve as initial conditions. For principal sl_2-subalgebras, the authors obtain explicit formulas, and for arbitrary partition types they describe the cases k=1 and k=2.

For an sl_2-subalgebra of type [d_1,d_2,…,d_m], the authors state that the general computation becomes difficult beyond k=2 and that the branching rules for k≥3 and arbitrary partitions cannot be expressed using the principal-subalgebra results alone. They identify the need for other techniques, leaving the explicit determination of these branching rules unresolved.