Characters and conjugacy classes of the symmetric group
Identifieur interne : 00B472 ( Main/Merge ); précédent : 00B471; suivant : 00B473Characters and conjugacy classes of the symmetric group
Auteurs : Alain Goupil [Canada] ; Dominique Poulalhon [France] ; Gilles Schaeffer [France]Source :
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Abstract
This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of $\S{n}$ viewed as operators acting by multiplication. The first conjecture expresses, for a fixed partition $\r$ of the form $r1^{n-r}$, the eigenvalues (or central characters) $\eo\r\l$ in terms of contents of $\l$. While Katriel conjectured a generic form and an algorithm to compute missing coefficients, we provide an explicit expression. The second conjecture (presented at FPSAC'98 in Toronto) gives a general form for the expression of a conjugacy class in terms of \emph{elementary} operators. We prove it using a convenient description by differential operators acting on symmetric polynomials. To conclude, we partially extend our results on $\eo\r\l$ to arbitrary partitions $\r$. || Nous démontrons plusieurs conjectures dues à Jacob Katriel qui portent sur les classes de conjugaisons de $\S{n}$ vues comme opérateurs agissant par multiplication. La première conjecture exprime, pour une partition fixée $\rho$ de la forme $r1^{n-r}$, le
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<profileDesc><textClass><keywords scheme="mix" xml:lang="fr"><term>caractères</term>
<term>characters</term>
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<term>constantes de structure</term>
<term>content</term>
<term>contenu</term>
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<front><div type="abstract" xml:lang="en">This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of $\S{n}$ viewed as operators acting by multiplication. The first conjecture expresses, for a fixed partition $\r$ of the form $r1^{n-r}$, the eigenvalues (or central characters) $\eo\r\l$ in terms of contents of $\l$. While Katriel conjectured a generic form and an algorithm to compute missing coefficients, we provide an explicit expression. The second conjecture (presented at FPSAC'98 in Toronto) gives a general form for the expression of a conjugacy class in terms of \emph{elementary} operators. We prove it using a convenient description by differential operators acting on symmetric polynomials. To conclude, we partially extend our results on $\eo\r\l$ to arbitrary partitions $\r$. || Nous démontrons plusieurs conjectures dues à Jacob Katriel qui portent sur les classes de conjugaisons de $\S{n}$ vues comme opérateurs agissant par multiplication. La première conjecture exprime, pour une partition fixée $\rho$ de la forme $r1^{n-r}$, le</div>
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