We prove an upper bound for characters of the symmetric groups. In particular, we show that there exists a constant $a>0$ with a property that for every Young diagram $\lambda$ with $n$ boxes, $r(\lambda)$ rows and $c(\lambda)$ columns $$ \left| \frac{\mathrm{Tr}\, \rho^{\lambda}(\pi)}{\mathrm{Tr}\, \rho^{\lambda}(e)} \right| \leq \left[a \max\left(\frac{r(\lambda)}{n},\frac{c(\lambda)}{n},\frac{|\pi|}{n} \right)\right]^{|\pi|}, $$ where $|\pi|$ is the minimal number of factors needed to write $\pi\in S_n$ as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov’s and Biane’s character formulas and give a new formula for free cumulants of the transition measure.
@article{10_4007_annals_2011_173_2_6,
author = {Valentin F\'eray and Piotr \'Sniady},
title = {Asymptotics of characters of symmetric groups related to {Stanley} character formula},
journal = {Annals of mathematics},
pages = {887--906},
year = {2011},
volume = {173},
number = {2},
doi = {10.4007/annals.2011.173.2.6},
mrnumber = {2776364},
zbl = {05960673},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.2.6/}
}
TY - JOUR AU - Valentin Féray AU - Piotr Śniady TI - Asymptotics of characters of symmetric groups related to Stanley character formula JO - Annals of mathematics PY - 2011 SP - 887 EP - 906 VL - 173 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.2.6/ DO - 10.4007/annals.2011.173.2.6 LA - en ID - 10_4007_annals_2011_173_2_6 ER -
%0 Journal Article %A Valentin Féray %A Piotr Śniady %T Asymptotics of characters of symmetric groups related to Stanley character formula %J Annals of mathematics %D 2011 %P 887-906 %V 173 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.2.6/ %R 10.4007/annals.2011.173.2.6 %G en %F 10_4007_annals_2011_173_2_6
Valentin Féray ; Piotr Śniady. Asymptotics of characters of symmetric groups related to Stanley character formula. Annals of mathematics, Tome 173 (2011) no. 2, pp. 887-906. doi: 10.4007/annals.2011.173.2.6
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