Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

We study asymptotics of an irreducible representation of the symmetric group SnSn corresponding to a balanced Young diagram λ   (a Young diagram with at most Cn rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that|χλ(π)|=|Trρλ(π)Trρλ(e)|⩽(Dmax(1,|π|2n)n)|π|, where |π||π| denotes the length of a permutation (the minimal number of factors necessary to write π as a product of transpositions). Our main tool is an analogue of the Frobenius character formula which holds true not only for cycles but for arbitrary permutations.