Representation Theory of Finite Groups

本研究的目的是将Artin环和阶的Auslander-Reiten理论应用于有限群的表示理论。首先，我们考虑了完全离散估值环上群环的Auslander-Reiten分量的形状，并证明了当群是素数幂阶时，那些包含射影格的分量的形式为ZA_<∞>。此外，我们还证明了平凡源格位于其Auslander-Reiten分量的末端。因此，我们得到了Heller-Reiner定理的另一个证明，该定理断言在完全离散估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运用对称群表示理论中的深层结果，得到了关于对称群Hecke代数的一些结果。此外，Asashiba还得到了一些等价条件，证明了树型分段遗传代数的扭曲多重扩展是等价的。此外，他还通过循环颤振代数的霍尔代数实现了一般线性代数和特殊线性代数。

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.