Upper bound on the characters of the symmetric groups

Let C be a conjugacy class in the symmetric group S_n, and λ be a partition of n. Let f^λ be the degree of the irreducible representation S^λ,_χ^λ(C) - the character of S^λ at C, and r^λ(C) - the normalized character χ^λ(C)/fλ. We prove that there exist constants b > 0 and 1 > q > 0 such that for n > 4, for every conjugacy class C in Sn and every irreducible representation S^λ of S_n (formula presented) where supp(C) is the number of non-fixed digits under the action of a permutation in C, λ₁ is the size of the largest part in λ, and λ′₁ is the number of parts in λ 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.