M-systems and Cluster algebras

TL;DR

This paper introduces M-systems and dual M-systems for types A and B, and establishes their connection to cluster algebras, proving the Hernandez-Leclerc conjecture for minimal affinizations in these types.

Contribution

It introduces new M-systems and dual M-systems and proves the Hernandez-Leclerc conjecture for specific algebraic structures.

Findings

Established the connection between M-systems and cluster algebras.

Proved the Hernandez-Leclerc conjecture for minimal affinizations of types A and B.

Introduced four new systems of equations for types A and B.

Abstract

The aim of this paper is two-fold: (1) introduce four systems of equations called M-systems and dual M-systems of types $A_{n}$ and $B_{n}$ respectively; (2) make a connection between M-systems (dual M-systems) and cluster algebras and prove that the Hernandez-Leclerc conjecture is true for minimal affinizations of types $A_n$ and $B_n$.