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.

Bratteli diagram of alternating groups

  • 1
    $\begingroup$ Thanks for the two references. I have downloaded them. The results are very important and interesting. $\endgroup$ – Xueyi Huang Oct 23 '18 at 7:18

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