Upper bound on the characters of the symmetric groups

Let C be a conjugacy class in the symmetric group S-n, and lambda be a partition of n. Let f(lambda) be the degree of the irreducible representation S-lambda, chi(lambda)(C) - the character of S-lambda at C, and r(lambda)(C) - the normalized character chi(lambda)(C)/f(lambda). We prove that there exist constants b>0 and 1>q>0 such that for n>4, for every conjugacy class C in S-n and every irreducible representation S-lambda of S-n \r(lambda)(C)\less than or equal to(max{q,lambda(1)/n,lambda(1)'/n})(b*supp(C)) where supp(C) is the number of non-fixed digits under the action of a permutation in C, lambda(1) is the size of the largest part in lambda, and lambda(1)', is the number of parts in lambda. The proof is obtained by enumeration of rim hook tableaux, the Hook formula and probabilistic arguments. Combinatorial, algebraic and statistical applications follow this result. In particular, we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes.