Character values at a cyclic permutation of a symmetric group

Question

Let $S_n$ be the symmetric group of degree $n$ and $\sigma\in S_n$ be a cyclic permutation of order $p$, where $p$ is a prime and $p>n/2$. Consider ordinary irreducible characters of $S_n$. Are there any bounds on values of the characters at $\sigma$? I am also interested in bounds on values of irreducible Brauer characters.

Answer 1

Here's a slightly modified version of Geoff's answer that doesn't use modular representation theory but just the representation theory of the symmetric group.

I'll assume the characters of $\mathfrak{S}_n$ are written as $\chi^{\alpha}$ with $\alpha \vdash n$ a partition of $n$. Assume $w \in \mathfrak{S}_n$ is a $p$-cycle then according to the Murnaghan--Nakayama rule we have

$$\chi^{\alpha}(w) = \sum_{h_{ij}^{\alpha} = p} (-1)^{l_{ij}^{\alpha}}\chi^{\alpha\setminus R_{ij}^{\alpha}}(1).$$

Here $h_{ij}^{\alpha}$ is the $(i,j)$-hook of $\alpha$, $l_{ij}^{\alpha}$ is the leg length of the hook and $R_{ij}^{\alpha}$ is the associated rim hook.

As $p > \frac{n}{2}$ we have $\alpha$ has at most one $p$-hook. Indeed, assume $\alpha$ has two $p$-hooks then $p^2$ divides $n!$ by the hook formula for $\chi^{\alpha}(1)$. Hence, there exist two terms $\frac{n}{2} < k\leqslant \ell \leqslant n$ divisible by $p$ because $p > \frac{n}{2}$. In particular, we have $p$ divides the difference $0 \leqslant \ell - k < \frac{n}{2}$, which is impossible unless $k = \ell$. By the constraints of the question we can assume $n \geqslant 4$ which means $p^2 > \frac{n^2}{4} \geqslant n$ can't divide $k \leqslant n$ so there is at most one $p$-hook.

Applying this to the above formula shows that we have $\chi^{\alpha}(w) = 0$ unless $\alpha$ has a $p$-hook $h_{ij}^{\alpha} = p$, in which case

$$\chi^{\alpha}(w) = (-1)^{l_{ij}^{\alpha}}\chi^{\alpha\setminus R_{ij}^{\alpha}}(1).$$

Here $\chi^{\alpha\setminus R_{ij}^{\alpha}}(1)$ is the degree of an irreducible character of $\mathfrak{S}_{n-p}$. Hence one would need to bound these degrees to bound the values at $w$.

Answer 2

I can't answer the question about the values of Brauer irreducible characters, (were you thinking about Brauer characters of $p$-modular representations, or other characteristics, by the way? I think both should be possible to treat.). Let me give an explanation of how the theory of blocks with cyclic defect group ( even of defect one in this case) can be applied here to determine the values taken at $\sigma$ by complex irreducible characters. There are much more general block-theoretic results available now for $S_{n},$ but the cyclic defect theory works well here. Let $G = S_{n}$ and $\sigma$ be a $p$-cycle in $G$. Since it is assumed that $n <2p,$ note that $\langle \sigma \rangle$ is a Sylow $p$-subgroup of $G.$ Let's relabel the points if necessary so that $\sigma$ fixes $i$ for $p+1 \leq i \leq n.$

Then $[N_{G}(\langle \sigma \rangle ): C_{G}(\langle \sigma \rangle)] = p-1,$ and, in fact, $\sigma$ is already to conjugate to all of its non-identity powers within $S_{p}.$ Setting $H = S_{p}$ (in its natural embedding in $G$) and considering $\sigma$ as an element of $H,$ we see that $N_{G}(\langle \sigma \rangle) \cong N_{H}(\langle \sigma \rangle) \times S_{n-p},$ where the rightmost symmetric group is acting on the fixed points of $\sigma.$ Also, with these identifications, we have $C_{G}(\sigma) \cong \langle \sigma \rangle \times S_{n-p}.$

Any irreducible character of $G$ which lies in a $p$-block whose defect group is not $\langle \sigma \rangle$ vanishes at $\sigma$ ( and these are precisely the irreducible characters of $G$ which have degree divisible by $p).$

By Brauer's first main theorem, there is a bijection between $p$-blocks of $G$ with defect group $\langle \sigma \rangle$ and $p$-blocks of $N_{G}(\langle \sigma \rangle )$ with defect group $\langle \sigma \rangle$. Since the latter group is a direct product of a Frobenius group of order $p(p-1)$ with $S_{n-p},$ it follows easily that the number of such blocks is the number of irreducible characters of $S_{n-p},$ which we denote (in standard fashion) by $k(S_{n-p})$ (which is, in this case, the number of partitions of $n-p).$

I won't give all details, but the structure of $N_{G}(\langle \sigma \rangle)$ and the theory of blocks of defect one now tell us in this situation that for each irreducible character $\mu$ of $S_{n-p},$ there are exactly $p$ irreducible characters of $G$, say $\{ \chi^{\mu}_{i} : 1 \leq i \leq p \}$ and $p$ associated signs $\epsilon_{i}$ such that for all $y \in S_{n-p}$ ( identified as before), and for each $i,$ we have $\chi^{\mu}_{i}(\sigma y) = \epsilon_{i}\mu(y).$ In particular, $\chi^{\mu}_{i}(\sigma) = \epsilon_{i}\mu(1).$

Every irreducible character $\chi$ of $G$ with $\chi(\sigma) \neq 0$ occurs once and only once as $\chi^{\mu}_{i}$ for some irreducible character $\mu$ of $S_{n-p}$ and sign $\epsilon_{i}.$

In particular, the number of complex irreducible characters of $S_{n}$ which do not vanish at $\sigma$ is $p\pi(n-p),$ where $\pi(m)$ is the number of partitions of the positive integer $m.$

Later edit: Using a well known upper bound for the largest irreducible character degree (which may be found stated, for example, in a 1975 paper of J.McKay),we see that $|\chi(\sigma)| \leq (2\pi)^{\frac{1}{4}}(n-p)^{\frac{1}{4}}\left( \frac{n-p}{e}\right)^{ \frac{n-p}{2}}$ for any complex irreducible character $\chi$ of $S_{n}$ if $n >p$ and when $n = p,$ we have $|\chi(\sigma)| \leq 1$ for all such $\chi.$

Answer 3

I will confine my comment to ordinary characters. I suspect that if $p$ is not too close to $n$ then the largest character value will be roughly of size $\sqrt{(n-p)!}$. Let $\mu$ be a partition of $n-p$ that maximizes $f^\mu$, the dimension of the character indexed by $\mu$, over all partitions $\mu\vdash n-p$. Since $\sum_{\lambda\vdash n-p}(f^\lambda)^2=(n-p)!$ and the number of partitions of $n-p$ is small compared to $\sqrt{(n-p)!}$, we have $f^\mu\approx \sqrt{(n-p)!}$. Now we should be able to perturb $\mu$ by a small amount turning it into a $p$-core $\nu$ (i.e., $\nu$ has no hook lengths equal to $p$). Add $p$ to the first (largest) part of $\nu$, obtaining a partition $\rho$. By the Murnaghan-Nakayama rule, the value of the character $\chi^\rho$ at a $p$-cycle is $f^\mu$. The same rule shows that the character value cannot be significantly larger even if $\rho$ has lots of hook lengths equal to $p$. (It certainly can't have more than $n$ of them.)