Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_{n1}\subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $A_n$ to $A_{n1}$? 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 GelfandTsetlin 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 nonselfconjugate 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 selfconjugate 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 $n1$, write $\mu\in \lambda^$ if the representation $V_\lambda$ of $S_n$ contains the representation $V_\mu$ of $S_{n1}$ upon restriction, then we have:
If $\lambda$ and $\mu$ are nonselfconjugate then the representation $V_\mu$ of $A_{n1}$ is contained in the restriction of $V_\lambda$ from $A_n$ to $A_{n1}$ if either $\mu\in \lambda^$, or $\mu'\in \lambda^$.
If $\lambda$ is nonselfconjugate and $\mu\in \lambda^$ is selfconjugate, then $V_\mu^\pm$ are both contained in the restriction of $V_\lambda$ from $A_n$ to $A_{n1}$.
If $\lambda$ is selfconjugate and $\mu\in \lambda^$ is nonselfconjugate, then $V_\mu$ is contained in the restriction of both $V_\lambda^\pm$ from $A_n$ to $A_{n1}$.
Finally, if $\lambda$ and $\mu\in \lambda^$ are both selfconjugate, 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 selfconjugate representation of $S_n$).
See the figure below. Please write to me if you would like an eprint of the the published version of the article.

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