We study variety algebraic objects by reflecting them into vector spaces. In particular, we use Young-object and Auslander-Retien quiver, which is very well-known combinaorial object, to define algebraic actions and study the phenomena happening by the actions to study those algebraic objects. We focus on quantum affine algebras and quiver Hecke algebra as the algebraic objects, which have origins from mathematical physics, by using the combinatorial tools.
We study the ring spanned by representations over variety algebraic objects, whose multiplication is a tensor indeed. This kind of study is established by Grothendieck, the french mathematian, around 1970's. In particular, we are interested in the relation between categorical representation theory and cluster algebra theory, both of them have natural positivity phenomena.