Asymptotics of characters of symmetric groups related to Stanley character formula | Annals of Mathematics

Abstract 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 λ with n boxes, r(λ) rows and c(λ) columns ∣∣∣Trρλ(π)Trρλ(e)∣∣∣≤[amax(r(λ)n,c(λ)n,|π|n)]|π|, where |π| is the minimal number of factors needed to write π∈Sn 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.

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数学年刊(Annals of Mathematics)

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