Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula

TL;DR

This paper derives explicit formulas for scalar products of Bethe vectors in models with $rak{gl}(2|1)$ symmetry, facilitating analysis of on-shell Bethe vectors and providing a determinant formula for $rak{gl}(1|1)$ models.

Contribution

It presents a novel sum-over-partitions representation for scalar products in $rak{gl}(2|1)$ models, extending the algebraic Bethe ansatz techniques.

Findings

Derived a sum over partitions formula for scalar products.

Obtained a determinant representation for $rak{gl}(1|1)$) models.

Facilitated analysis of on-shell Bethe vectors.

Abstract

We study scalar products of Bethe vectors in integrable models solvable by nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(2|1)$ symmetry. Using explicit formulas of the monodromy matrix entries multiple actions onto Bethe vectors we obtain a representation for the scalar product in the most general case. This explicit representation appears to be a sum over partitions of the Bethe parameters. It can be used for the analysis of scalar products involving on-shell Bethe vectors. As a by-product, we obtain a determinant representation for the scalar products of generic Bethe vectors in integrable models with $\mathfrak{gl}(1|1)$ symmetry.