Representation Theory and its Applications
Representation theory is a field whose purpose is to develop techniques for finding matrices or linear operators that satisfy a given relation in a systematic way. Among group, ring and field, which are introduced in standard abstract algebra courses, rings and fields are commutative algebraic structures, while groups are noncommutative ones. Hence, very roughly speaking, representation theory deals with noncommutative entities, while algebraic geometry and number theory deal with commutative entities. (In carrying out real research, we use these intertwined, depending on the nature of the problem under investigation.) There was a research group (often called "Sato school") in the Research Institute for Mathematical Sciences in Kyoto, which advocated algebraic analysis. They succeeded in handling soliton theory and theory of solvable lattice models that appear in mathematical physics using representation theory. Using this physical representation theory as a motif, the central theme in Ariki laboratory is the study of finite dimensional algebras arising in Lie theory, algebraic groups and quantum groups. We are thrilled with the way that multiple methods intertwine, and the way that quantities that are primarily very hard to compute can be simplified down to the level of combinatorics and may become computable, by applying the methods we develop. In recent years we have also begun to consider representation theory in Geometric Complexity Theory (GCT), which is a subject in theoretical computer science.
- Apr. 1989-Mar. 2002 Associate Professor, Dept. of Logistics and Information Engineering, Faculty of Marine Technology, Tokyo University of Marine Science and Technology
- Apr. 2002-Mar. 2010 Associate Professor, Research Institute for Mathematical Sciences, Kyoto University
- Apr. 2010- Current Position
- 2003 Awarded with the Autumn Prize of The Mathematical Society of Japan.
- 2011- Member of the Steering Comittee of the Algebra Section, The Mathematical Society of Japan.
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