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

Abstract

We study asymptotics of an irreducible representation of the symmetric group S n corresponding to a balanced Young diagram λ (a Young diagram with at most C n 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 | χ λ ( π ) | = | Tr ρ λ ( π ) Tr ρ λ ( e ) | ⩽ ( D max ( 1 , | π | 2 n ) n ) | π | , where | 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.