A recursion formula for Branching from ${\mathfrak{s}\mathfrak{l}}_n$ to ${\mathfrak{s}\mathfrak{l}}_2$ subalgebras

For any representation of a complex simple Lie algebra ${\mathfrak{s}\mathfrak{l}}_n$, one problem of branching rules to ${\mathfrak{s}\mathfrak{l}}_2$-subalgebra is to determine the multiplicity of each irreducible component. In this paper, we derive a recursion formula of such multiplicities by restricting a certain tensor representation in two ways, in which the Pieri’s rule is involved. We also investigate branching rules for fundamental representations as they are initial conditions of the recursion formula.