Quantum groups were introduced independently by Drinfeld
and Jimbo around 1985. Representation theory of quantum
groups has been developed intensively in the past decades
by many famous mathematicians. Many problems in
representation theory of quantum groups have been solved
but many fundamental problems are still open.
In this project, we study representations of quantum groups
and its connection with Grassmannian cluster algebras,
Grassmannian cluster categories and scattering amplitudes
in physics. We consider the following problems.
1. Study geometric cactus group actions on geometric
crystals. 2. In BCFG types, define an algebra homomorphism
D from the torus containing q-characters of simple modules
of quantum affine algebras, and apply D to study relation
between q-characters of representations of quantum affine
algebras and characters of representations of KLR algebras.
3. Study a tropical version of braid group actions and
Marsh-Scott twists on Grassmannian cluster algebras
C[Gr(k,n)] in terms of g-vectors and tableaux. Compare
Fraser`s braid group actions and Kashiwara, Kim, Oh, and
Park`s braid group actions. 4. Make a connection between
additive categorifications of Grassmannian cluster algebras
and monoidal categorification of Grassmannian cluster
algebras using g-vectors. Try to prove that rigid
indecomposable modules (resp. real prime modules) are
cluster variables using Iyama-Yoshino reduction. 5.
Applications to physics: study factorizations of
Grassmannian string integrals.