Decomposition of the conjugacy representation of the symmetric groups.

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Consider the two natural representations of the symmetric groupS n on the group algebra ℂ[S n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS λ in the conjugacy representation and letf λ be the multiplicity ofS λ in the regular representation. By the character estimates of [R1] and [Wa] we prove (1) For any 1>ε>0 there exist 0<δ(ε) andN(ε) such that, for any partitionλ ofn>N(ε) with max {λ1n,λ′1n}⩽δ(ε),, 1−ε<m(λ)fλ<1+ε whereλ 1 is the size of the largest part inλ andλ′1 is the number of parts inλ. (2) For any fixed 1>r>0 and ε>0 there existκ=κ(ε, r) andN(ε, r) such that, for any partitionλ ofn>N(ε, r) with max{λ1n,λ′1n}<r,, A−ε<m(λ)fλ<A+ε whereA is a constant which depends only on the fractions λ1n,...,λ1n,λ′1n,...,λ′kn. This strengthens Adin-Frumkin's result [AF] and answers a question of Stanley [St].
Original languageAmerican English
Pages (from-to)305-316
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - 1997


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