# Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$

## TL;DR

This paper introduces a novel matrix product solution to the reflection equation linked to a coideal subalgebra of quantum affine algebra, utilizing $q$-hypergeometric series and connecting to crystal base theory at $q=0$.

## Contribution

It provides a new matrix product formula for the reflection equation solution associated with $U_q(A^{(1)}_{n-1})$ coideal subalgebras, expanding the understanding of integrable systems.

## Key findings

- Matrix product formula involving $q$-hypergeometric series
- Solution reduces to a known crystal base solution at $q=0$
- Applicable to symmetric tensor representations and their duals

## Abstract

We present a new solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.