An identity in the Bethe subalgebra of $\mathbb{C}[\mathfrak{S}_n]$

TL;DR

This paper presents a new algebraic identity in the symmetric group algebra that simplifies the understanding of the Gaudin model's eigenspaces, bypassing the Bethe ansatz method.

Contribution

It introduces a novel identity in the Bethe subalgebra of the symmetric group algebra, enabling direct proof of the correspondence without Bethe ansatz techniques.

Findings

Established a new algebraic identity in the symmetric group algebra.

Provided a direct proof of the correspondence between inverse Wronskians and eigenspaces.

Simplified the proof of the Bethe ansatz conjecture for the Gaudin model.

Abstract

As part of the proof of the Bethe ansatz conjecture for the Gaudin model for $\mathfrak{gl}_n$, Mukhin, Tarasov, and Varchenko described a correspondence between inverse Wronskians of polynomials and eigenspaces of the Gaudin Hamiltonians. Notably, this correspondence afforded the first proof of the Shapiro-Shapiro conjecture. In the present paper, we give an identity in the group algebra of the symmetric group, which allows one to establish the correspondence directly, without using the Bethe ansatz.