# Branching Rule for alternating groups

Let $$A_n$$ be the alternating group of degree $$n$$. What is the branching rule for the subgroup $$A_{n-1}\subset A_n$$, i.e., the structure of the restriction of ordinary irreducible representations of $$A_n$$ to $$A_{n-1}$$? Are there some nice books or references which provide detailed answer to this question?

This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, and is also available on the arXiv. See also Ruff, O.: Weight theory for alternating groups. Algebra Colloq. 15(03), 391–404 (2008).

The alternating groups have two types of representations, those corresponding to mutually conjugate non-self-conjugate pairs $$(\lambda,\lambda')$$ which are simply the restriction of the irreducible representation $$V_\lambda$$ or $$V_{\lambda'}$$ of $$S_n$$ to $$A_n$$, and two corresponding to each self-conjugate partition $$\lambda$$, denoted $$V_\lambda^\pm$$, which are the two irreducible summands of the irreducible representation $$V_\lambda$$ of $$S_n$$, when resticted to $$A_n$$.

If $$\lambda$$ is a partition of $$n$$ and $$\mu$$ is a partition of $$n-1$$, write $$\mu\in \lambda^-$$ if the representation $$V_\lambda$$ of $$S_n$$ contains the representation $$V_\mu$$ of $$S_{n-1}$$ upon restriction, then we have:

1. If $$\lambda$$ and $$\mu$$ are non-self-conjugate then the representation $$V_\mu$$ of $$A_{n-1}$$ is contained in the restriction of $$V_\lambda$$ from $$A_n$$ to $$A_{n-1}$$ if either $$\mu\in \lambda^-$$, or $$\mu'\in \lambda^-$$.

2. If $$\lambda$$ is non-self-conjugate and $$\mu\in \lambda^-$$ is self-conjugate, then $$V_\mu^\pm$$ are both contained in the restriction of $$V_\lambda$$ from $$A_n$$ to $$A_{n-1}$$.

3. If $$\lambda$$ is self-conjugate and $$\mu\in \lambda^-$$ is non-self-conjugate, then $$V_\mu$$ is contained in the restriction of both $$V_\lambda^\pm$$ from $$A_n$$ to $$A_{n-1}$$.

4. Finally, if $$\lambda$$ and $$\mu\in \lambda^-$$ are both self-conjugate, then $$V_\mu^+$$ is contained in $$V_\lambda^+$$ and $$V_\mu^-$$ is contained in $$V_\lambda^-$$. This result is based on a careful choice of sign in defining the representations $$V_\lambda^\pm$$ (deciding which gets the $$+$$ sign, and which gets the $$-$$ minus sing among the irreducible $$A_n$$ representations contained in a self-conjugate representation of $$S_n$$).

See the figure below. Please write to me if you would like an e-print of the the published version of the article.

• Thanks for the two references. I have downloaded them. The results are very important and interesting. – Xueyi Huang Oct 23 '18 at 7:18