Journal of Lie Theory 23 (2013), No. 3, 691--697
Copyright Heldermann Verlag 2013
A Remark on Pillen's Theorem for Projective Indecomposable kG(n)-Modules
National College of Technology, 22 Yata, Yamatokoriyama, Nara, Japan 639-1080
Let $g$ be a connected, semisimple and simply connected algebraic group defined and split over the finite field of order $p$, and let $g(n)$ be the corresponding finite chevalley group and $g_n$ the $n$-th frobenius kernel. Pillen has proved that for a $3(h-1)$-deep and $p^n$-restricted weight $\lambda$, the $G$-module $Q_n(\lambda)$ which is extended from the $G_n$-PIM for $\lambda$ has the same socle series as the corresponding $kG(n)$-PIM $U_n(\lambda)$. Here we remark that this fact already holds for $\lambda$ being $2(h-1)$-deep.
Keywords: Loewy series, projective indecomposable modules, 2(h-1)-deep weights
MSC: 20C33, 20G05, 20G15
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