Consider the elements of the group algebra CS_{ n} given by R_{ j}=Sigma_{ i=1}^{ j-1}(ij), for 2<=j<=n. Jucys [3 - 5] and Murphy[7] showed that these elements act diagonally on elements of S_{ n} and gave explicit formulas for the diagonal entries. As requested by the late S. Kerov, we give a combinatorial proof of this work in case j=n and present several similar results which arise from these combinatorial methods.
Categories: math.CO math.RT