A Bethe Ansatz for Symmetric Groups

We examine the commuting elements $\theta_i=\sum_{j\neq i} \frac{s_{ij}}{z_i-z_j}$,$z_i\neq z_j$,$s_{ij}$thetranspositionswapping$i$and$j$,andwestudytheiractionsonirreducible$S_n$ representations. By applying Schur-Weyl duality to the results of \cite{RV:QuasiKZ}, we establish a Bethe Ansatz for these operators which yields joint eigenvectors for each critical point of a master function. By examining the asymptotics of the critical points, we establish a combinatorial description (up to monodromy) of the critical points and show that, generically, the Bethe vectors span the irreducible $S_n$ representations.