Piecewise Linear Representation Theory of Quantum Groups and Geometric Crystals

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The theory of geometric crystal is obtained as an analogus theory on algebraic varieties to crystal bases by considering certain group actions on the vatirety, which turns out to be an analogus operator to the crystal operators. It is well-known that by the tropicalization / ultra-discretization procedure we obtain crystals from geometric crystals.The purpose of the research is to construct a geometric crystal structure on various algebraic varieties. In fact, I have succeeded to construct the geometric crystal structure on the Schubert varieties associated with Kac-Moody groups. Furthermore, I have constructed the geometric crystal structures on the affine Schubert varieties which is obtained from the translations of the extended Weyl groups. This geometric crystal possesses the following remarkable properties : it has a natural positive structure and the associated crystal is isomorphic to the so-called the limit of perfect crystals. We obtained the tropical R maps on the product of … More these geometric crystals, which is an analogus object to R-matrices. Perfect crystals are crucial objects in the study of affine type crystals and they play a central role in the theory of solvable lattice models in mathematical physics and the theory of Kirillov-Reshetikhin modules.As for the representation of affine quantum groups t roots of 1, we obtain the sufficient and necessary condition for that two evaluation representations are isomorphic to each other.The Markov trace is one of important topological invariants. From the view point of topology and representation theory, the Markov trace turns out to be an rather interesting object which we should study. In order to treat the Markov trace in the general framework, Gomi defined the Markov property and present the way to construct the Markov trace by the unified method.Shinoda succeeded in obtaining certain interesting results on relations between zeta functions and the Gel'fand Graev representations of finite reductive groups by performing the explicit calculations. Less