Upper bound on the characters of the symmetric groups
Upper bound on the characters of the symmetric groups
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对称群特征的上限
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发表时间：
1996
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通讯作者：
Yuval Roichman
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作者：
Yuval Roichman
设C是对称群Sn中的一个共轭类，λ是n的一个划分。设fλ为不可约表示Sλ的阶，设χλ(C)是Sλ at C的特征，设rλ(C)是正则化特征χλ(C) fλ。我们证明存在常数b > q > 0和1 > q > 0，使得对于n > 4，对于Sn中的每一个共轭类C和Sn的每一个不可约表示Sλ∣rλ(C)∣≦(max{q,λ 1n,λ1 ' n})b·supp(C)
Abstract 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∣rλ(C)∣≦ (max{q,λ1 n, λ1′ n})b ⋅ supp(C)
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 λ1′ 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.