My Research

I'm interested in crystal bases in a representation theory of Lie algebras or quantum groups. 

The quantum group is a q-analog of universal enveloping algebras of Lie algebras g. 

The crystal bases are, roughly speaking, bases of representations of quantum groups at q = 0, which are 

powerful combinatorial tools to reveal structures of representations of Lie algebras and quantum groups.

Crystal bases have a bunch of combinatorial descriptions like as tableaux realizations, path realizations, 

geometric realizations, etc. I mainly treat the polyhedral realizations, which realize the crystal bases as integer points of

some polyhedral cones or polytopes. 

The following is an example of polyhedral realization for crystal base B(∞) of Verma module in case of g = sl_3(C) : 

Here, (0,0,0) is called the highest weight vector and f_1 and f_2 mean action of operators called Kashiwara operators. 

The set of vectors (a_3,a_2,a_1) appearing in the above graph coincides with the following set of integer points in a polyhedral cone:

The following is the most fundamental problem in the theory of polyhedral realizations:

"Find explicit forms of polyhedral cones or polytopes like as above"

I'm working on related topics to this problem:

1. When g is an affine Lie algebra, we describe inequalities that define the polyhedral cones and polytopes in terms of combinatorial objects like as Young walls, extended diagrams and so on. We also describe the inequalities in terms of monomial realizations and give a representation theoretical interpretation.

2. When g is a finite dimensional simple Lie algebra, the polyhedral cone coincides with string cones. In the context of geometric crystals, by computing Berenstein-Kazhdan decorations, one obtains an explicit form of the cone. We give an algorithm to compute the BK decorations.

3. We'd like to generalize the algorithm in 2. to calculations of Gross-Hacking-Keel-Kontsevich potentials on double Bruhat cells. It appears in the context of cluster algebras.