Branching of Representations to Symmetric Subgroups

TL;DR

This paper provides explicit algorithms and LiE programs for computing the branching of finite-dimensional representations of compact Lie algebras when restricted to symmetric subalgebras defined by automorphisms of various orders.

Contribution

It introduces a systematic method and software implementation for explicit branching calculations for automorphisms of orders 2, 3, 5, and centralizers of toral subalgebras.

Findings

Explicit branching rules for automorphisms of orders 2, 3, and 5.

LiE programs for practical computation of representation restrictions.

Applications to symmetric and nearly-Kähler homogeneous spaces.

Abstract

Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex representation $\pi$ of $\gg$, give the explicit branching $\pi|_\gk$ of $\pi$ on $\gk$. Cases of special interest include the cases where $\theta$ has order 2 (corresponding to compact riemannian symmetric spaces $G/K$), where $\theta$ has order 3 (corresponding to compact nearly--kaehler homogeneous spaces $G/K$), where $\theta$ has order 5 (which include the fascinating 5--symmetric space $E_8/A_4A_4$), and the cases where $\gk$ is the centralizer of a toral subalgebra of $\gg$.