Upper bound on the characters of the symmetric groups

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Let C be a conjugacy class in the symmetric group Sn, and λ be a partition of n. Let fλ be the degree of the irreducible representation Sλ,χλ(C) - the character of Sλ at C, and rλ(C) - the normalized character χλ(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 Sλ of Sn (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.

Original languageEnglish
Pages (from-to)451-485
Number of pages35
JournalInventiones Mathematicae
Issue number3
StatePublished - Jul 1996
Externally publishedYes


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