Decomposition of the conjugacy representation of the symmetric groups

Abstract

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 $\{ \frac{{\lambda _1 }} {n},\frac{{\lambda '_1 }} {n}\} \leqslant \delta \left( \varepsilon \right),$,

  $$1 - \varepsilon< \frac{{m\left( \lambda \right)}}{{f^\lambda }}< 1 + \varepsilon $$

  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$\{ \frac{{\lambda _1 }} {n},\frac{{\lambda '_1 }} {n}\} < r,$,

  $$A - \varepsilon< \frac{{m\left( \lambda \right)}}{{f^\lambda }}< A + \varepsilon $$

  whereA is a constant which depends only on the fractions

  $$ \frac{{\lambda _1 }} {n},...,\frac{{\lambda _1 }} {n},\frac{{\lambda '_1 }} {n},...,\frac{{\lambda '_k }} {n}. $$

  This strengthens Adin-Frumkin’s result [AF] and answers a question of Stanley [St].