Representation Theory of Finite Groups

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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.