Asymptotics of characters of symmetric groups related to Stanley character formula. (English) Zbl 1229.05276

Summary: 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.


05E10 Combinatorial aspects of representation theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20C30 Representations of finite symmetric groups
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