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

Korkeat Korkeathikhun, Borworn Khuhirun, Songpon Sriwongsa, Keng Wiboonton

TL;DR

The paper develops a recursion for multiplicities in the branching of $\mathfrak{sl}_n$-modules to an $\mathfrak{sl}_2$-subalgebra. It derives the main formula $m_d(\lambda+\omega_k)=\sum_{(j,j')} m_j(\lambda)m_{j'}(\omega_k)-\sum_{\mu\in P(\lambda,k)\setminus\{\lambda+\omega_k\}}m_d(\mu)$, with a parity and triangle-inequality constraint on $(j,j')$, and the special case $m_0(\lambda+\omega_k)=\sum_j m_j(\lambda)m_j(\omega_k)-\sum_{\mu\in P(\lambda,k)\setminus\{\lambda+\omega_k\}}m_0(\mu)$. Fundamental representations provide the initial data, while principal subalgebras yield explicit multiplicities via partitions and plethysms (e.g., $m_j(\omega_k)= p_k^n(\frac{kn-j+k}{2})-p_k^n(\frac{kn-j+k}{2}-1)$) and Hermite reciprocity; non‑principal cases are treated in special shapes with explicit decompositions and illustrative examples. The work connects branching problems to combinatorics of partitions and plethysms, offering a practical computational tool for computing restriction multiplicities with potential applications in representation theory and combinatorics.

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.

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

TL;DR

The paper develops a recursion for multiplicities in the branching of -modules to an -subalgebra. It derives the main formula , with a parity and triangle-inequality constraint on , and the special case . Fundamental representations provide the initial data, while principal subalgebras yield explicit multiplicities via partitions and plethysms (e.g., ) and Hermite reciprocity; non‑principal cases are treated in special shapes with explicit decompositions and illustrative examples. The work connects branching problems to combinatorics of partitions and plethysms, offering a practical computational tool for computing restriction multiplicities with potential applications in representation theory and combinatorics.

Abstract

For any representation of a complex simple Lie algebra , one problem of branching rules to -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.

Paper Structure

This paper contains 6 sections, 6 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Recursion Formula for Multiplicity of $F_d$
  3. Branching rules of Fundamental $\mathfrak{sl}_n$ Representations
  4. Principal subalgebras
  5. Non-principal subalgebras
  6. Examples

Key Result

Theorem 1.1

Let $\mathfrak{s}$ be an $\mathfrak{sl}_2$-subalgebra of $\mathfrak{sl}_n$. For each dominant integral weight $\lambda\in \Lambda^+$, let $m_j(\lambda)=\mathrm{mult}(F_{j}: \mathrm{Res}_{\mathfrak{s}}^{\mathfrak{sl}_n}L(\lambda))$. Then for $d\in \mathbb{Z}_{\geq 0}$ and $k=1, 2, \dots, n-1$, where the first summation ranges over $(j,j')\in \mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$ such that

Theorems & Definitions (19)

  • Theorem 1.1
  • Example 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • ...and 9 more