Consider the following multiple contour integral:

$$ \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j - z_k) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n z_j^{1-j - \lambda_j} dz_1 \ldots dz_n.$$

This gives a representation of the Schur polynomial $s_\lambda(x_1, \ldots, x_n)$ according to my preliminary calculation, based on the homogenous symmetric polynomial version of the Jacobi-Trudi identity. Similar formulae seem to exist under the guise of generalized hypergeometric series, which can be defined via Jack symmetric polynomial. These can be viewed as generalization of Selberg integrals, via the conformal map $z = \frac{i + y}{i-y}$.

I am interested in the case where $x_i \in \mathbb{T}:= \{z \in \mathbb{C}: |z| = 1\}$ for all $i$. Such quantities arise naturally as irreducible characters of $U(n)$ for instance.

Even more restrictively, I am interested in the case where the centroid of the $x_i$'s is zero, that is, $\sum_{j=1}^n x_j = 0$. Thus define $V := \{(x_1, \ldots, x_n) \in \mathbb{T}^n: \sum_j x_j = 0\}$.

One way of generating such $x_i$'s is as a mixture of extremal configurations of the form $\{e^{\alpha_k + \frac{2\pi i j}{m_k}}: 1 \le j \le n_k\}$, such that $m_k \mid n_k$ and $\sum_k n_k = n$. As a special case, if $n$ is even, we can match any $n/2$ points on $\mathbb{T}$ with their antipodal points to satisfy the constraint. But there are many other ways to do it, as long as $n > 4$.

My question is whether this leads to an effective estimate of the quantity $$ s^*_{\lambda, V} := \max_{(x_1, \ldots, x_n) \in V} |s_\lambda(x_1, \ldots, x_n)|.$$

In particular, my conjecture is that

$$s^*_{\lambda, V} \le \sqrt{s_\lambda(1, \ldots, 1)},$$ provided $\lambda \neq (k,\ldots, k)$ for some $k \ge 0$.

Fully credit will be given if one can show for some positive $\epsilon$,

$$s^*_{\lambda, V} \le s_\lambda(1,\ldots, 1)^{1 - \epsilon + o(1)},$$ under the same condition on $\lambda$.

The reason the conjecture should fail for such fully rectangular $\lambda$ is that the random walk on $U(n)$ generated by a conjugacy class from $V$ always lives on a single left-coset under the quotient by $SU(n)$. The fully rectangular representations correspond to the other irreducible components of the walk, whose traces are nonzero. I have actually managed to prove the conjecture with an asymptotically negligible error term when $\lambda$ consists of one row or one column only, that is, when $s_\lambda$ is either elementary or homogeneous, based on a convexity argument. In those cases the Vandermonde factor $\prod_{1 \le j < k \le n}(z_j - z_k)$ drops out, so the analysis is much easier.

I would be happy if someone can present a proof of the weaker conjecture in the case $\lambda = (kn, k(n-1), \ldots, k)$. For $k = 2$, and $n$ even, we can compute $s_\lambda(-1,1,\ldots, (-1)^j, \ldots, 1) = \sqrt{s_\lambda(1, \ldots, 1)}$, because (by Macdonald section 1.4 example 1) $$s_\lambda(x_1, \ldots, x_n) = \prod_{1 \le j < k \le n} (x_j^2 + x_j x_k + x_k^2),$$ and about half of the time $x_j, x_k$ have opposite signs, which contributes $1$ to the product. Same argument can be made for any even $k > 0$. For odd $k$, this yields $0$. This does not prove the conjecture for such $\lambda$'s, but the appearance of square root was encouraging.

Going back to the original multiple integral, in the special case $\lambda = (n,n-1, \ldots, 1)$. One way to estimate its norm is by putting absolute value around all factors, restricting all $z_j$'s to a circle of radius $r$, and estimating the maximal value the integrand can attain: $$ \Psi(\vec{x},r) := \max_{\vec{\theta}} \prod_{1 \le j < k \le n} (r | e^{i \theta_j} - e^{i \theta_k}|) \prod_{j=1}^n \prod_{k=1}^n |1 - e^{i \theta_j} x_k|^{-1} r^{-n^2}.$$

For fixed $\vec{x}$ and $r$, $\Psi(\vec{x}, r)$ is related to the weighted logarithmic capacity on the unit circle, with the weight function given by $w(z) = -\sum_{k=1}^n \log| 1 - r z x_k|$.

Does the configuration $x_j = (-1)^j$ always maximize $\Phi(., r)$? It seems that for $r \approx 1$, $x_j = e^{2\pi i j/ 3}$ gives a higher value, which reminds me of the optimality competition between binary and ternary number systems. So a related question is whether $\Phi(., r)$ is maximized at an $\vec{x}$ of the form $x_j = e^{2 \pi i j / k}$ for some $k \mid n$.

Any partial result/insight/reference/comment is very welcome.