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I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$.

Symmetric group $S_n$

Due to Hardy and Ramanujan (1918), $n$ can be partitioned in approximately $$ p(n) \sim \frac{1}{4 n \sqrt{3}} e^{\pi\sqrt{\frac{2 n}{3}}} $$ ways.

It means the number of conjugacy classes hence the number of irreps of $S_n$ is approximately $\frac{1}{4 n \sqrt{3}} e^{\pi\sqrt{\frac{2 n}{3}}}$.

Dihedral group $D_n$ of order $2 n$

For $D_n$, the one dimensional representations are as follows.

When $n$ is even:

  • The trivial representation, sending all group elements to the $1 \times 1$ matrix $\begin{pmatrix}1\end{pmatrix}$.
  • The representation, sending all elements in $\langle x \rangle$ to $\begin{pmatrix}1\end{pmatrix}$ and all elements outside $\langle x \rangle$ to $\begin{pmatrix}-1\end{pmatrix}$.
  • The representation, sending all elements in $\langle x^2, y \rangle$ to $\begin{pmatrix}1\end{pmatrix}$ and $x$ to $\begin{pmatrix}-1\end{pmatrix}$.
  • The representation, sending all elements in $\langle x^2, x y \rangle$ to $\begin{pmatrix}1\end{pmatrix}$ and $x$ to $\begin{pmatrix}-1\end{pmatrix}$.

When $n$ is odd:

  • The trivial representation, sending all group elements to the $1 \times 1$ matrix $\begin{pmatrix}1\end{pmatrix}$.
  • The representation, sending all elements in $\langle x \rangle$ to $\begin{pmatrix}1\end{pmatrix}$ and all elements outside $\langle x \rangle$ to $\begin{pmatrix}-1\end{pmatrix}$.

Moreover, when $n$ is even, the number two dimensional irreps is $\frac{n-2}{2}$. When $n$ is odd, it is $\frac{n-1}{2}$.

Here, we list the $k$-th two dimensional irreducible representations for the general group elements.

\begin{align} x \mapsto \begin{pmatrix} e^{\frac{2 \pi i k}{n}}&0\\ 0&e^{-\frac{2 \pi i k}{n}} \end{pmatrix} \nonumber\\ x^l \mapsto \begin{pmatrix} e^{\frac{2 \pi i k l}{n}}&0\\ 0&e^{-\frac{2 \pi i k l}{n}} \end{pmatrix} \nonumber\\ y \mapsto \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \nonumber\\ x^l y \mapsto \begin{pmatrix} 0&e^{\frac{2 \pi i k l}{n}}\\ e^{-\frac{2 \pi i k l}{n}}&0 \end{pmatrix} \end{align}

So,when $n$ is even, the total number of irreps is $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. When $n$ is odd, it is $\frac{n-1}{2} + 2 = \frac{n+3}{2}$.

My question:

How do I restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$? In other words, what is $Res^{S_n}_{D_n}$ for all irreps of $S_n$?

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  • $\begingroup$ I am following the definition given in the section 1.12 of [The Symmetric Group : Representations, Combinatorial Algorithms, and Symmetric Functions][1]. The definition is given as follows. Consider $H \le G$ and a matrix representation $X$ of $G$. The restriction of $X$ to $H$, $X\downarrow^G_H $, is given by $$X\downarrow^G_H (h) = X(h)$$ for all $h \in H$. [1]: link.springer.com/book/10.1007%2F978-1-4757-6804-6 $\endgroup$ – Omar Shehab May 12 '16 at 4:29
  • $\begingroup$ So, by definition, for a given irrep $\rho$ of $S_n$, we will pick any element $h$ from $H$, and the set of all $\rho (h)$ will be $\rho$'s restriction from $G$ to $H$. $\endgroup$ – Omar Shehab May 12 '16 at 4:43

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