Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by 
\begin{equation}
s\_\lambda(x\_1,\ldots, x\_n) = a\_{\lambda + \delta}(x\_1, \ldots, x\_n) / a\_\delta (x\_1, \ldots, x\_n),
\end{equation}
 where $\delta = (n-1,n-2,\ldots, 0)$ and $a\_\lambda = \det (x\_i^{\lambda\_j})$ is the Vandermonde determinant of the $n \times n$ matrix whose $(i,j)$ element is $x\_i^{\lambda\_j}$. My notation here is consistent with Ian Macdonald's Hall polynomial book chapter 1.


Also let $e\_j$ be the $j$th elementary symmetric polynomial in $n$ variables, $0 \le j \le n$. These are defined by 
\begin{equation}
\prod\_{i=1}^n (1 + x\_i t) = \sum\_{j=0}^n e\_j(x\_1,\ldots, x\_n) t^j.
\end{equation}
By the Jacobi-Trudi formula we know that 
\begin{equation}
s\_\lambda = \det(e\_{\lambda^t\_i -i + j}),
\end{equation}
 where $\lambda^t$ is the transpose of the partition $\lambda$, that is, $\lambda^t\_i = \mid\{j: \lambda\_j \geq i\}\mid$. Thus $s\_{1^j} = e\_j$, for $j \le n$. 


Now we specialize to $x\_i \in \mathbb{T}:= \{z \in \mathbb{C}: \mid z \mid = 1\}$. Given that
\begin{equation}
e\_1(x\_1, \ldots, x\_n) = 0,
\end{equation}
 that is, $\sum x\_i = 0$, I am interested in bounding $s\_\lambda$. 


I have a conjecture for $\lambda = 1^{n/2}$, and $n = 4m$, $m \in \mathbb{N}$, namely, 
\begin{equation}
\mid e\_{n/2}(x\_1, \ldots, x\_n) \mid \le \binom{n/2}{n/4}.
\end{equation}
This is attained when $x\_i = (-1)^i$, that is, when half of them equal $1$ and the other half equal $-1$, since 
\begin{equation}
\prod\_i (1 +x\_i t) = (1-t)^{n/2} (1+t)^{n/2} = (1-t^2)^{n/2}.
\end{equation} 


I don't know if this conjecture is true for all qualifying $x\_i$'s. Full credit will be given to solve this special case. However, I am also interested in a general conjecture for arbitrary $\lambda$.



Bounding Schur symmetric polynomials on the unit circle
In the special case mentioned in the problem, I'll show the bound 
$$ 
|e\_{n/2}(x\_1,\ldots, x\_n)| \le 2^{n/2}. 
$$ 
Let 
$$ 
F(z) = \prod\_{j=1}^{n} (1+zx\_j) = C \prod\_{j=1}^{n} (z + \overline{x\_j}), 
$$ 
where $C= \prod\_{j} x\_j$ has magnitude $1$. (The roots of $F$ are $-\overline{x\_j}$.)


Now by Cauchy's theorem 
$$ 
e\_{n/2}(x\_1,\ldots,x\_n) = \frac{1}{2\pi i} \int\_{|z|=1} \frac{F(z)}{z^{n/2+1}} dz 
$$ 
and so in magnitude this is bounded by 
$$ 
\le \sup\_{|z|=1} |F(z)|. 
$$ 


Now we use a very nice Theorem of [Carneiro and Vaaler](http://arxiv.org/pdf/0809.4050.pdf) (see Theorem 8.1 there), which gives a bound for the maximum of a polynomial whose roots are on the unit circle in terms of the first few symmetric power sums of the roots. Let me quote their result fully: Suppose $G(z) = \prod\_{j=1}^{N}(z-\alpha\_j)$ is a polynomial with $|\alpha\_j| \le 1$ for all $j$. Then for any natural number $K$ we have 
$$ 
\sup\_{|z|\le 1} \log |G(z)| \le \frac{N}{K+1} \log 2 + \sum\_{k=1}^{K} \frac{1}{k} \Big| \sum\_{j=1}^{N} \alpha\_j^k \Big|.
$$ 
This is Theorem 8.1, display (8.6) of their paper. 


Apply this to our polynomial $F$, with $K=1$. Since the sum of the $x\_j$ is zero, by assumption, we deduce that 
$$ 
\max\_{|z|=1} \log |F(z)| \le \frac{N}{2} \log 2, 
$$ 
which proves the claimed estimate. 


Maybe there is an easier proof of this particular bound, rather than appealing to the Carneiro-Vaaler work. I didn't see one immediately, and would be very interested if someone found an alternative approach. 


Edit: In the particular case $K=1$, which is what is needed for this question, [zeb](https://mathoverflow.net/users/2363/zeb) kindly showed me the following one line proof: assuming $|\alpha\_j|\le 1$ and $|z|\le 1$ we have 
$$
|G(z)|^2 =\prod\_{j} |z-\alpha\_j|^2 \le \prod\_{j} (2-2\text{Re }\overline{z}\alpha\_j) \le 2^N \prod\_j \exp(-\text{Re }\overline{z}\alpha\_j)\le 2^N \exp\Big(\Big| \sum\_j \alpha\_j\Big|\Big).
$$ 


