On Euclidean Components of Auslander-Reiten Quivers of Finite Group Schemes

Abstract

We are interested in components of the stable Auslander-Reiten quiver ${\Gamma }_{s}(\mathcal {G})$ of a finite group scheme $\mathcal {G}$ over a field k of characteristic p. By definition, ${\Gamma }_{s}(\mathcal {G})$ is defined to be the stable Auslander-Reiten quiver of the finite-dimensional cocommutative Hopf algebra $k\mathcal {G}:=k[\mathcal {G}]$, the dual Hopf algebra of its coordinate ring $k[\mathcal {G}]$. Every component Θ of ${\Gamma }_{s}(\mathcal {G})$ gives rise to a tree $\overline {T}_{\Theta }$. If p ≥ 3, will show that the presence of a non-trivial unipotent normal subgroup $\mathcal {N}\unlhd \mathcal {G}$ implies that $\overline {T}_{\Theta }$ is not Euclidean and that Θ is not isomorphic to $\mathbb {Z}[\tilde {A}_{p,q}]$ for any natural numbers p,q. This will be proven by using certain invariants of components defined by p-points of $\mathcal {G}$. In case of a finite group G, which can be seen as a special instance of a finite group scheme, we will then obtain an alternative proof of Okuyama’s theorem, which asserts that the abovementioned components will in general not occur in that case. Further applications will be given in the contexts of trigonalizable group schemes, Frobenius kernels G_r of (reductive) algebraic groups G and for finite group schemes $\mathcal {G}$ with tame principal block ${\mathscr{B}}_{0}(\mathcal {G})$.