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 Sn corresponding to a balanced Young diagram λ (a Young diagram with at most C sqrt(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| χλ (π) | = | frac(Tr ρλ (π), Tr ρλ (e)) | ≤ (frac(D max (1, frac(| π |2, n)), sqrt(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. © 2008 Elsevier Inc. All rights reserved.