2003 Fiscal Year Final Research Report Summary
Combinatorial Representation Theory of Quantum Groups
| Project/Area Number |
13640043
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Sophia University |
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Principal Investigator |
NAKASHIMA Toshiki Sophia University, Department of Mathematics, Ass.Professor, 理工学部, 助教授 (60243193)
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| Co-Investigator(Kenkyū-buntansha) |
GOMI Yasushi Sophia University, Department of Mathematics, Assistant, 理工学部, 助手 (50276515)
YOKONUMA Takeo Sophia University, Department Mathematics, Professor, 理工学部, 教授 (00053645)
SHINODA Ken-ichi Sophia University, Department Mathematics, Professor, 理工学部, 教授 (20053712)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Quantum Groups / Crystal Bases / Braid Groups / Polyhedral Realizations / Maximal Cyclic Representations / coinvariant algebra / Hecke algebra / Gauss sum |
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| Research Abstract |
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
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