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

TL;DR

This paper derives a recursion formula for calculating multiplicities in the branching rules from l_n to l_2 subalgebras, using tensor restrictions and Pieri's rule, with applications to fundamental representations.

Contribution

It introduces a new recursion formula for branching multiplicities from l_n to l_2, involving tensor restrictions and Pieri's rule, and studies fundamental representations.

Findings

Derived a recursion formula for multiplicities

Connected branching rules to Pieri's rule

Analyzed fundamental representations as initial conditions

Abstract

For any representation of a complex simple Lie algebra $\mathfrak{sl}_n$, one problem of branching rules to $\mathfrak{sl}_2$-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.