A Bethe Ansatz for Symmetric Groups

Abstract

We examine the commuting elements ${\theta }_i={\sum }_{j\ne i}\frac{s_{ij}}{z_i-z_j}$, $z_i\ne z_j$, $s_{ij}$ the transposition swapping $i$ and $j$, and we study their actions on irreducible $S_n$ 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.

11 pages

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