A Bethe Ansatz for Symmetric Groups

TL;DR

This paper develops a Bethe Ansatz framework for symmetric group representations using commuting elements and Schur-Weyl duality, providing a combinatorial description of critical points and showing the Bethe vectors span the representations.

Contribution

It introduces a Bethe Ansatz for symmetric groups' operators, linking critical points of a master function to eigenvectors and their combinatorial structure.

Findings

Bethe vectors span irreducible $S_n$ representations.

A combinatorial description of critical points is established.

The approach connects algebraic operators with geometric critical point analysis.

Abstract

We examine the commuting elements $\theta_i=\sum_{j\neq i} \frac{s_{ij}}{z_i-z_j}$, $z_i\neq z_j$, $s_{ij}$ 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.