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Define $T^d$ as following $$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\} $$

For any partition $\lambda\vdash n$,The Schur function is defined

$$ \displaystyle s_\lambda(x_1, \ldots, x_d) = \frac{\det\Bigl(x_i^{d + \lambda_j -j}\Bigr)_{ij}}{\det\Bigl(x_i^{d-j}\Bigr)_{ij}}. $$

I would like to ask the (upper and lower) bounds in $u$ of $$ |s_\lambda(t)| $$ given that $|t_i-t_j|<u$ for all $i,j$.

I am also interested in bounding $|s_\lambda(t)|$ if there exists $i,j$ such that $|t_i-t_j|>u$

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  • $\begingroup$ Hopefully someone else will chime in with a more complete answer/reference, but just a first observation that lower bounds are more interesting in this setting than upper-bounds since no matter how small u is, under this constraint, the maximum of $s_\lambda(t)$ will be equal to $s_\lambda(1^d)$, by examining the combinatorial definition of Schur functions (see Stanley, Enumerative Combinatorics vol. II, 7.10). Getting good estimates for $s_\lambda(1^d)$ can sometimes be done using the hook-length formula (see e.g. Corollary 7.21.4 of Stanley). $\endgroup$ Mar 8, 2019 at 22:40

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