Snake Modules And Cluster Algebras With Applications

Dynkin diagram is an important object in the field of mathematics.It is well known that Dynkin diagrams give a classification of complex semisimple Lie algebras,(crystal)root systems,Coxeter groups,finite representation type of quivers,cluster algebras of finite type,reflection monoids.With Dynkin diagram used as a bridge,we mainly study the connections between the cluster algebra theory and quantum affine algebra,as well as the cluster algebra theory and reflection monoids.In 2010,two French mathematicians Hernandez and leclerc introduced the con-cept of monoidal categorification of cluster algebras.They first studied the connection between cluster algebras and the finite dimensional representations of quantum affine algebras,and proved that there is a cluster algebra structure on the Grothedick ring of some subcategories of the category of finite dimensional representations of quantum affine algebras and conjectured(?)there is a one-to-one correspondence between cluster monomials and the iso-morphism classes of real simple objects;(?)there is a one-to-one correspondence between cluster variables and the isomor-phism classes of real and prime simple objects.(?)the normal truncated q-character of any real simple object equals to the F-polynomial of some rigid module(not unique)over Jacobian algebra(defined by quiver with potentials).We prove that the Hernandez-Leclerc Conjecture holds for a class of special mod-ules called snake modules by Mukhin and Young.In 2015,Barot and Marsh studied the presentation of finite irreducible crystal reflection groups by the method of cluster algebras.We study the connection toward to two different directions generalization.More specifically,as a generalization of the ordinary quivers,we consider some quivers with forzen vertices;as a generalization of reflection groups,we consider reflection monoids introduced by Everitt and Fountain.We show that the inverse monoids defined by Dynkin diagrams with "frozen" vertices are isomorphic to Everitt and Fountain's Boolean reflected monoids.We construct the graph inner automorphism of irreducible Weyl group and Boolean reflection monoid by the mutation sequence preserving the same underlying diagram.Moreover,we prove the Boolean reflection monoid algebra is a cellular algebra.