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Central Characters and Conjugacy Classes in the Symmetric Group

Abstract : This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of $\S{n}$. 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 $\r$.
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https://hal.inria.fr/inria-00099194
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Submitted on : Tuesday, September 26, 2006 - 8:51:38 AM
Last modification on : Friday, February 26, 2021 - 3:28:02 PM

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Alain Goupil, Dominique Poulalhon, Gilles Schaeffer. Central Characters and Conjugacy Classes in the Symmetric Group. 12th International Conference on Formal Power Series and Algebraic Combinatorics - FPSAC'00, 2000, Moscou/Russia, pp.238-249. ⟨inria-00099194⟩

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