New symmetries of $\mathfrak{gl}(N)$-invariant Bethe vectors

TL;DR

This paper reveals a symmetry between two types of Bethe vectors in $rak{gl}(N)$-invariant quantum integrable models, showing they are essentially the same up to normalization, leading to new combinatorial relations.

Contribution

It establishes a novel symmetry between Bethe vectors associated with the monodromy matrix and its inverse, using the current approach.

Findings

Bethe vectors of two types are identical up to normalization.

New combinatorial relations for scalar products of Bethe vectors.

Symmetry simplifies analysis of $rak{gl}(N)$-invariant models.

Abstract

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(N)$-invariant $R$-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors.