Characters of symmetric groups in terms of free cumulants and Frobenius coordinates

TL;DR

This paper provides explicit combinatorial formulas for normalized characters of symmetric groups using free cumulants and Frobenius coordinates, linking algebraic and combinatorial aspects of Young diagrams.

Contribution

It introduces new explicit formulas connecting symmetric group characters with free cumulants and Frobenius coordinates, utilizing Stanley polynomials and permutation factorizations.

Findings

Explicit combinatorial formulas for normalized characters

Connection between free cumulants and Frobenius coordinates

Use of Stanley polynomials for character evaluation

Abstract

Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups S(n) in the limit n\to\infty. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams. R\'esum\'e. Les cumulants libres sont des fonctions agr\'eables et utiles sur l'ensemble des diagrammes de Young, en particulier, ils donnent le comportement asymptotiques des caract\`eres normalis\'es du groupe sym\'etrique S(n) dans la limite n\to\infty. Nous donnons une formule combinatoire explicite pour les caract\`eres normalis\'es du groupe sym\'etrique en fonction des cumulants libres. Nous exprimons \'egalement les caract\`eres en fonction des coordonn\'ees de Frobenius. Nos formules font intervenir le nombre de certaines factorisations d'une permutation donn\'ee. L'outil principal est la famille de polyn\^omes de Stanley donnant les valeurs des caract\`eres sur les diagrammes de Young multirectangulaires.