Central Characters and Conjugacy Classes of the Symmetric Group or On some Conjectures of J. Katriel

Abstract

This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of S n viewed as operators acting by multiplication. The first one expresses, for a fixed partition ρ of the form r1 n−r , the eigenvalues (or central characters) ω ρ λ in terms of contents of λ. 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 elementary operators. We prove it using a convenient description by differential operators acting on symmetric polynomials. To conclude, we partially extend our results on ω ρ λ to arbitrary partitions ρ.