representations of $S_k \times S_j$

Question

For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$). I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible.

Answer 1

I guess that $S_n$ denotes the symmetric group. It is well known that all irreps of $S_n$ over $\mathbb C$ are defined over $\mathbb Q$. Therefore $S(S_k\times S_{n-k},\mathbb{C})=S(S_k\times S_{n-k},\mathbb{Q})$. Also the fact that the sum of the squares of the dimensions is the group order and Cauchy-Schwarz immediately imply that $S(G,\mathbb C)\le \sqrt{c(G)|G|}$ where $c(G)$ is the number the conjugacy classes. In your case it means $S(S_n,\mathbb C)\le \sqrt{p(n)n!}$ where $p$ is the partition function.

Answer 2

Since, as mentioned by Friedrich Knop, the irreducible $\mathbb{Q}$-representations of $S_{n}$ are absolutely irreducible, ( so, in particular all complex irreducible characters of $S_{n}$ are real valued) it follows that $\sum_{\chi \in {\rm Irr}(S_{n})}\chi(1)$ is the number of solutions of $x^{2} = 1$, where $\chi$ runs over the complex irreducible characters of $S_{n}$, which is the same in this case as the sum of the degrees of irreducible rational representations of $S_{n}$. This uses the well-known theory of the Frobenius-Schur indicator.

Hence the sum in question is $ 1+ \sum_{k=1}^{\lfloor \frac{n}{2} \rfloor} \frac{n!}{(n-2k)! 2^{k}k!}$, since the centralizer of a product of $k$ disjoint transpositions is isomorphic to $ S_{n-2k} \times (S_{2} \wr S_{k})$.

The same reasoning applies to direct products of symmetric groups.

If you wish to approximate sums such as those above, the middle terms tend to dominate ($k$ close to $\frac{n}{2}$), and just taking $ k = \frac{n}{2}$ if $n$ is even or $k \in \{\frac{n \pm 1}{2}\}$ if $n$ is odd can be used to directly obtain a bound of the form $p(n) \sim e^{c \sqrt{n}}$ for the partition function.

Answer 3

If all representations of a group can be realized over $\mathbb{R}$, then $S(G, \mathbb{C})$ equals the number of elements of order 2 in $G$. More generally, the number of solutions of the equation $x^2=g$ in $G$ is $\sum\epsilon_\chi \chi(g)$, where $\epsilon_\chi=1$, if $\chi$ belongs to a real representation, $\epsilon_\chi=-1$, if $\chi$ has non-real values, and $\epsilon_\chi=0$, if all values of $\chi$ are real, but the representation associated to $\chi$ is not real.

Chowla, Herstein and Moore have shown that the number of solutions of $x^2=1$ in $S_n$ is asamptotically equal to $$ \frac{n^{n/2}e^{\sqrt{n}}}{\sqrt{2}e^{n/2+1/4}}, $$ Müller has shown that there exists an asymptotic expansion in terms of $n^{-1/2}$.