Combinatorial Representation Theory of Quantum Groups

量子群的组合表示论

• 批准号: 13640043
• 负责人: NAKASHIMA Toshiki
• 金额: $ 1.98万
• 依托单位: Sophia University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640043/
• 关键词: Quantum Groups; Crystal Bases; Braid Groups; Polyhedral Realizations; Maximal Cyclic Representations; coinvariant algebra; Hecke algebra; Gauss sum; 表現論; 極外ベクトル; 極大巡回加群; ゼータ関数; イプシロン因子; 組合せ論; 1の巾根

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

我们学习了量子群的表示理论和相关的组合数学。特别是，首席研究员通过多面体实现的方法研究了晶体基础理论。多面体实现是将一个晶体基嵌入到某个无限整数格中，并将其实现为某个凸多面体上的格点。它的特点是使我们能够明确地计算许多事情。事实上，对于几种类型的量子群，他成功地给出了晶体基和极值向量的明确形式。他还研究了量子群的根单位。至于最大循环表示的A型，专门化其参数适当，他给不可约模块的限制量子代数。此外，他比较了它与无穷小Verma模。对于Chevalley群G_2（q），给出了7次模表示和11次幂单不可约表示的Gauss和.在证明的过程中，他还提出了几种求和公式。Gomi利用组合方法，特别是Reidemeister-Schreier定理，给出了与Coxeter群相关的纯辫群的一些表示

项目成果

Y.Gomi, I.Nakamura, K.Shinoda: "Coinvariant Algebras of finite subgroups of SL(3,C)"Canadian Journal of Mathematics. (to appear).

Y.Gomi、I.Nakamura、K.Shinoda：“SL(3,C) 有限子群的协变代数”加拿大数学杂志。

中島俊樹: "Polytopes for Crystallized Demazure modules and Extremol Vectors"Communications in Algebra. 30.3. 1349-1367 (2002)

Toshiki Nakajima：“结晶 Demazure 模和极值向量的多面体”代数通讯 30.3（2002）。

K.Shinoda, C.W.Curtis: "Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group"Advanced Studies in Pure Mathematics. (to appear).

K.Shinoda、C.W.Curtis：“与有限还原群的 Gelfand-Graev 表示的分量相关的 Zeta 函数和函数方程”纯数学高级研究。

中島俊樹: "On certain maximal cyclic modules for the quantized special linear algebras at a root of unity"Pacific Journal. (in press).

Toshiki Nakajima：“关于统一根处的量化特殊线性代数的某些最大循环模”太平洋杂志（正在出版）。

Y.Gomi, F.Digne: "Presentation of pure braid groups"Journal of Knot Theory Ramifications. 10,4. 609-623 (2001)

Y.Gomi，F.Digne：“纯辫子群的呈现”结理论分支杂志。