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Decomposition of the conjugacy representation of the symmetric groups

Authors:

Yuval Roichman

Institution:

(1) Department of Mathematics, Harvard University, 02138 Cambridge, MA, USA

Abstract:

Consider the two natural representations of the symmetric groupS _n on the group algebra ℂS _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].

Partially sponsored by a Wolfson fellowship and the Hebrew University of Jerusalem.

Keywords:

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