Combinatorial Representation Theory of Quantum Groups

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We studied representation theory of quantum groups and related combinatorics. In particular, the head investigator researched theory of crystal bases by the method of polyhedral realizations. Polyhedral realization is to embed a crystal base in certain infinite integer lattice and realized it as a lattice points in some convex polyhedron. Its feature is to enable us to calculate many things explicitly. Indeed, for several types of quantum groups, he succeeded in giving explicit form of crystal bases and extremal vectors. He also studied quantum groups at roots of unity. As for maximal cyclic representations of type A, specializing its parameters properly, he gave irreducible modules of restricted quantum algebra. Furthermore, he compared it with infinitesimal Verma modules.K.Shinoda studied Chevalley groups and Hecke algebras. For Chevalley group G_2(q), we gave Gauss sums for modular representation of degree 7 and 11 unipotent irreducible representations. In the course of its proof, he also presented several kinds of summation formulae. In particular, he showed that on some elements, character values coincide with general Kloosterman sum.Y.Gomi gave some presentations of pure braid groups associated with Coxeter groups by combinatorial methods, in particular, by Reidemeister-Schreier theorem