Representation Theory of Finite Groups

有限群表示论

• 批准号: 13640038
• 负责人: KAWATA Shigeto
• 金额: $ 1.79万
• 依托单位: Osaka City 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-13640038/
• 关键词: FINITE GROUP; REPRESENTATION; AUSLANDER-REITEN QUIVER; GROUP RING; ALGEBRA; AUSLANDER-REITEN SEQUENCE; INTEGRAL REPRESENTATION; INDECOMPOSABLE MODULE; 群環

The purpose of this research project is to apply the Auslander-Reiten theory for Artin rings and orders to the representation theory of finite groups. First, we considered the shapes of the Auslander-Reiten components of group rings over complete discrete valuation rings and showed that those components containing the projective lattices are of the form ZA_<∞> if the groups are of prime power order. Moreover, we showed that trivial source lattices lie at the ends of their Auslander-Reiten components. Consequently, we obtained another proof of a theorem of Heller-Reiner, which asserts that the group rings of finite groups of prime cubed order over complete discrete valuation rings are of infinite representation type.On the other hand, Tsushima got some results concerning on the Hecke algebras of symmetric groups by applying the deep results in the representation theory of the symmetric groups.Also, Asashiba obtained some equivalent conditions which implies that twisted multifold extensions of piecewise hereditary algebras of tree type are derived equivalent. Furthermore, he realized general and special linear algebras via Hall algebras of cyclic quiver algebras.

本研究项目的目的是将Artin环和阶的Auslander-Reiten理论应用于有限群的表示理论。首先，我们研究了完全离散赋值环上群环的Auslander-Reiten分支的形状，并证明了如果群是素幂阶群，则包含投射格的那些分支具有ZA_<∞>的形式。此外，我们还证明了平凡源格位于它们的Auslander-Reiten分量的端点。由此，我们得到了Heller-Reiner定理的另一个证明，即完备离散赋值环上的素数立方阶有限群的群环是无限表示型的.另一方面，Tsushima利用对称群的表示论中的深入结果，得到了关于对称群的Hecke代数的一些结果. Asashiba得到了树型分片遗传代数的扭多重扩张是等价的等价条件。此外，他通过循环代数的Hall代数实现了一般和特殊线性代数。

项目成果

Hideto Asashiba: "On a lift of an individual stable eqivalence to a standard derived equivalence for representation-finite self-infective algebras"Algebras and Representation Theory. 6. 427-447 (2003)

Hideto Asashiba：“将个体稳定等价提升到表示有限自感染代数的标准导出等价”代数和表示理论。

A.Jones, S.Kawata, G.Michler: "On exponents and Auslander-Reiten components of irreducible lattices"Arch.Math. 76・2. 91-94 (2001)

A.Jones、S.Kawata、G.Michler：“关于不可约格子的指数和 Auslander-Reiten 分量”Arch.Math 76・2（2001）。

A.Jones, S.Kawata, G.O.Michler: "On exponents and Auslander-Reiten components of irreducible lattices"Archiv der Mathematik. 76. 91-94 (2001)

A.Jones、S.Kawata、G.O.Michler：“论不可约格子的指数和 Auslander-Reiten 分量”Archiv der Mathematik。

Yukio Tsushima: "On some SR(H)-blocks for the symmetric groups"Journal of Algebra. 270. 281-287 (2003)

Yukio Tsushima：“关于对称群的一些 SR(H) 块”代数杂志。

S.Kawata, G.Michler, K.Uno: "On Auslander-Reiten components and simple modules for finite groups of Lie type"Osaka Journal of Mathematics. 38. 21-26 (2001)

S.Kawata、G.Michler、K.Uno：“论李型有限群的 Auslander-Reiten 分量和简单模”《大阪数学杂志》。