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 $\lambda$.