On factorizing $F$-matrices in $Y(sl_n)$ and $U_q(\hat{sl_n})$ spin chains

TL;DR

This paper derives a fully factorized expression for the $F$-matrix in quantum spin chains based on $Y(sl_n)$ and $U_q(\\hat{sl_n})$ algebras, connecting it to Bethe eigenvectors and confirming previous results.

Contribution

It introduces a new factorized form of the $F$-matrix for these models and establishes its equivalence with prior expressions, also relating it to Bethe eigenvectors.

Findings

Derived a completely factorized $F$-matrix expression.

Proved equivalence with existing $F$-matrix formulas.

Established a new relation between $F$-matrices and Bethe eigenvectors.

Abstract

We consider quantum spin chains arising from $N$-fold tensor products of the fundamental evaluation representations of $Y(sl_n)$ and $U_q(\hat{sl_n})$. Using the partial $F$-matrix formalism from the seminal work of Maillet and Sanchez de Santos, we derive a completely factorized expression for the $F$-matrix of such models and prove its equivalence to the expression obtained by Albert, Boos, Flume and Ruhlig. A new relation between the $F$-matrices and the Bethe eigenvectors of these spin chains is given.